Layered graph models and exact algorithms for the generalized hop-constrained minimum spanning tree problem

Computers & Operations Research 65 (2016) 1–18

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Layered graph models and exact algorithms for the generalized hopconstrained minimum spanning tree problem Markus Leitner a,b,n a b

Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria Department of Computer Science, Université Libre de Bruxelles, Boulevard du Triomphe CP 210/01, 1050 Brussels, Belgium

art ic l e i nf o

a b s t r a c t

Available online 2 July 2015

This paper studies the generalized hop-constrained minimum spanning tree problem (GHMSTP) which has applications in backbone network design subject to quality-of-service constraints that restrict the maximum number of intermediate routers along each communication path. Different possibilities to model the GHMSTP as an integer linear program and strengthening valid inequalities are studied. The obtained formulations are compared theoretically, i.e., by means of their linear programming relaxation. In addition, branch-and-cut approaches based on these formulations are developed and compared in a computational study. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Integer programming Branch-and-cut Generalized network design Hop-constraints Layered graph

1. Introduction Generalized network design problems (GNDPs) with applications in backbone network design have been studied in detail in the recent years (see, e.g., [5,6,17,26] and the references therein) and exact as well as heuristic solution approaches for them have been proposed. Given an undirected graph G ¼ ðV; EÞ with nonnegative edge costs ce, 8 eA E, a GNDP is characterized by the fact that the node set V is partitioned into a set of K disjoint clusters Vk, 1 rk rK, V ¼ ⋃Kk ¼ 1 V k , V k \ V l ¼ ∅, 1 r k a l r K. A feasible solution S ¼ ðP; TÞ, P  V, T  E, is a subgraph of G which selects precisely one node from each cluster and connects these nodes according to the requirements of the concrete GNDP at hand, e.g., by a spanning tree in case of the generalized minimum spanning tree problem (GMSTP), cf. [6,8,25,27], or a Hamiltonian cycle in the generalized traveling salesman problem (GTSP), cf. [7,20,31]. The usual objective is to identify a P solution yielding minimum edge costs e A T ce . Applications of GNDPs arise for example in the design of backbone networks. Thereby, clusters represent areas (e.g., cities) each of which is connected via a local network and nodes model possibilities to interconnect these local networks. In case, failure tolerance by means of installing redundant connections does not need to be considered this application domain naturally leads to aforementioned GMSTP for which a variety of exact and heuristic solution approaches have been proposed in the recent scientific literature, see, e.g., [6,17] and the references therein.

n Correspondence address: Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.cor.2015.06.012 0305-0548/& 2015 Elsevier Ltd. All rights reserved.

In this work, we study a new variant of the GMSTP which additionally incorporates side constraints that allow to limit the communication delay of the resulting network in centralized networks. Several studies have suggested that reducing the number of intermediate routers or edges along a communication path increases availability and reliability of a network as well as the quality of the transmitted signal [22,33], see also [4]. It has also been observed that such bounds reduce the possibility of failures without the need to consider quite costly alternatives such as node- or edge-disjoint networks. Therefore, bounds on the total communication delay are typically ensured via enforcing an upper bound on the number of hops from a predefined distribution node to each other node (in centralized networks) or between any two nodes (i.e., a bound on the diameter of the resulting network) in decentralized networks. Consequently, in particular the hop-constrained minimum spanning tree problem (HMSTP) has been studied exhaustively in the scientific literature, see, e.g., [4,10,11,13,14,32]. The current state-of-the-art approach has been proposed by Gouveia et al. [14] who suggested a branch-and-cut algorithm based on a transformation of the HMSTP to a Steiner arborescence problem on an extended, so-called layered graph a concept which will be used intensively in this paper. The Generalized Hop-Constrained Minimum Spanning Tree Problem (GHMSTP) captures aforementioned aspects for the design of centralized backbone networks and is formally defined as follows. Let G ¼ ðV; EÞ be an undirected graph with costs ce Z 0 associated to each edge eA E whose node set V is partitioned into K mutually disjoint clusters ∅ aV k  V, 1 rk rK, ⋃Kk ¼ 1 V k ¼ V, V k \ V l ¼ ∅, 1 r k a l r K. Let furthermore without loss of generality V1 be the root cluster and H A A be the given hop limit. A feasible solution is a cycle-free, connected subgraph S ¼ ðP; TÞ, P  V, T  E, of G spanning

2

M. Leitner / Computers & Operations Research 65 (2016) 1–18

exactly one node from each cluster (i.e., P ¼ fv1 ; v2 ; …; vK g, vk A V k , 1 r k r K, that satisfies all hop constraints, i.e., the unique path between the chosen root node v1 A V 1 and any other chosen node consists of at most H edges. The objective is to identify a feasible P solution Sn ¼ ðP n ; T n Þ yielding minimum costs e A T n ce . An exemplary instance together with a feasible solution for H Z 3 is given in Fig. 1. The GHMSTP is NP-hard in general since it contains both the hop-constrained minimum spanning tree problem (see, [3,9]) as well as the GMSTP (see, [24]) as the special case when j V i j ¼ 1, i ¼ 1; …; K, and when H Z K  1, respectively. We note that for H¼ 1 the problem can, however, be solved in polynomial time by the following algorithm: First select a root node u A V 1 . Then, for each remaining cluster Vi, 2 ri r K, choose a node v A V i that can be connected to the root u at minimum cost, i.e., v ¼ argminw A V i cuw and add the edge fu; vg to the partial solution. By iterating over all possible root nodes u A V 1 and adopting the overall cheapest solution, one obtains an optimal solution to the considered instance of the GHMSTP for H¼ 1. Scientific contribution: Besides introducing the new problem, the main results of this paper are (a) Integer linear programming (ILP) models (presented in Sections 2 and 3) that successfully combine ideas from related generalized network design problems with those from hop-constrained network design problems; (b) A theoretical study comparing all proposed formulations by means of their linear programming relaxation values in Section 4; (c) The development and computational comparison of branch-and-cut approaches based on these formulations in Section 5. Our results show that a variant based on the concept of a layered cluster graph (see Section 3) outperforms all other considered algorithms. Note, that this paper focuses on layered graph formulations in which connectivity is ensured by directed connectivity constraints. The formulations proposed here can, however, easily be modified to use (multi-commodity) flows instead. Flow-based formulations have been used in the literature with great success for many network design problems with hop- or diameter constraints, see, e.g., [1,10,12]. We do, however, refrain from explicitly describing flow-based formulations, as branch-and-cut approaches based on efficiently separable connectivity constraints (such as the ones considered in the remainder of this paper) are in many cases preferable from a computational perspective. Notation: In all formulations considered in the following, a solution to the GHMSTP is represented as an outgoing arborescence rooted at a node from V1 which spans all selected nodes. Therefore, we utilize arc set A ¼ fðu; vÞj fu; vg A E; u A V 1 g [ fðu; vÞj fu; vg A E; u; v2 = V 1 g. For each edge fu; vg adjacent to a node u A V 1 an arc (u,v) emanating from u is considered while two oppositely directed arcs are added for all remaining edges. Given a set of variables f defined on all members P of set S, and a subset S0 D S, we will use notation f ½S0  ¼ i A S0 f i . Furthermore, for subsets U  V and W  V we will denote by δðU; WÞ ¼ fðu; wÞ A Aj u A U; w A Wg the set of arcs from U to W and  by δ ðWÞ ¼ fðu; vÞ A Aj u2 = W; v A Wg the ingoing cutset for W.

Fig. 1. An illustrative instance of the GHMSTP with six clusters and an exemplary feasible solution S ¼ ðP; TÞ with P ¼ fv1 ; …; v6 g with vi A V i , 1 r ir 6, for H Z 3. Edges not contained in the solution are not shown.

Analogous notation will be used to denote ingoing cutsets for subsets of nodes of graphs different from G. Finally, we will use C ¼ f1; …; Kg to denote the set of clusters and VðwÞ ¼ fu A V i j w A V i g to refer to all nodes that are contained in the same cluster than w A V. 2. Layered graph models The formulations introduced in this section make use of layered graph GL ¼ ðV L ; AL Þ which encodes the distance of nodes and arcs from the chosen root node into its structure. Node set VL contains copies u0 of all nodes u from the root cluster V1 at layer zero and copies uh of all other nodes u A V⧹V 1 on layers 1 r h r H. Arcs ðuh ; vh þ 1 Þ are included in arc set AL whenever an arc (u,v) exists in arc set A. Thus, GL which is formally defined by V L ¼ fu0 j u A V 1 g [ fuh j u A V⧹V 1 ; 1 r h r Hg and AL ¼ fðuh ; vh þ 1 Þj ðu; vÞ A Ag is acyclic and does not contain paths longer than the given hop limit H. Fig. 2 shows the layered graph representation of the solution given in Fig. 1 for H ¼3. We note that, in particular for sparse input graphs and when using an appropriate construction or preprocessing routine, node set VL is typically not complete (i.e., some nodes may not exist at all layers). To simplify notation, we nevertheless assume a complete node set in the following. Formulations (1)–(8) to which we will refer to as Lx make use of node decision variables zi A f0; 1g, 8 i A V, indicating whether or not a node is selected as well as arc design variables xuv A f0; 1g, 8 ðu; vÞ A A, denoting whether or not arc (u,v) is contained in the directed solution. Furthermore, layered arc design variables X huv A f0; 1g, 8 ðuh ; vh þ 1 Þ A AL , 0 r h o H, are used to define the positions of each used arc in the arborescence, i.e., X huv ¼ 1 if and only if arc (u,v) is used and node u is at distance h from the root node. X min cuv xuv ð1Þ ðu;vÞ A A

s:t:

z½V i  ¼ 1

8iAC



x½δ ðuÞ ¼ zu 

x½δ ðWÞ Zzu xuv þ xvu r zu H 1 X

X huv ¼ xuv

ð2Þ

8 u A V⧹V 1

ð3Þ

8 W  V⧹V 1 ; u A W 8 u A V; 8 fu; vg A E 8 ðu; vÞ A A

ð4Þ ð5Þ ð6Þ

h¼0

X ðuh  1 ;vh Þ A AL ;u 2 = Vi

X huv 1 Z

X

X hvw

8 vh A V L ; 8 i A C; h Z1; δðfvg; V i Þ a ∅

ðvh ;wh þ 1 Þ A AL ;w A V i

ð7Þ ðx; z; XÞ A f0; 1g j Aj þ j V j þ j AL j

ð8Þ

The objective function (1) minimizes the costs of the obtained solution while Eq. (2) ensure that precisely one node from each cluster will be selected. Indegree constraints (3) link arc and node

Fig. 2. Embedding of the solution given in Fig. 1 on layered graph GL for H¼3. Arcs not part of the solution are not shown and nodes not selected are not labeled.

M. Leitner / Computers & Operations Research 65 (2016) 1–18

variables and ensure that precisely one ingoing arc is selected for each chosen node that is not contained in the root cluster. Inequalities (4) are directed connectivity constraints that enforce connectivity of the solution. Together with inequalities (5) – which ensure that the two nodes adjacent to a selected arc are selected as well – they ensure that there exists a path from the selected root node to each other selected node. Observe, that (1)–(5) resembles a model for the GMSTP previously used by Myung et al. [24] as well as by Feremans [5]. Equations (6) link layered arc to arc design variables. Together with (3) they also ensure that for each arc ðu; vÞ A A at most one of its layered copies ðuh ; vh þ 1 Þ can be used and that the corresponding target node j has to be selected as well. Finally, inequalities (7) make sure that an arc emanating from node vh, 1 r h r H, to a node in cluster i may only be used if an ingoing arc originating at a cluster different from i is used as well. Since GL is acyclic this polynomial number of constraints ensure connectivity of the solution obtained on the layered graph, cf. [30]. Though inequalities (7) are sufficient to ensure connectivity, it is well known from similar layered graph reformulations [14,29] that one can strengthen such a formulation by additionally considering connectivity cuts (9) defined on GL. We will use Lcx to refer to the stronger variant of model Lx augmented by these constraints. 

X½δ ðWÞ Z zi

8 W D V L ⧹fu0 : u A V 1 g; fih ; 1 r h r Hg D W

ð9Þ

We also observe that connectivity constraints (4) can be strengthened if all nodes from at least one cluster are contained in set W, yielding 

x½δ ðWÞ Z 1

8 W D V⧹V 1 ; ( k A C⧹f1g : V k D W

ð10Þ

to which we will refer to as cluster cuts while (4) will be called node cuts. Their validity can be easily seen since a path from the selected root node to the node selected in cluster k must exist in every feasible solution. The models additionally containing (10) þ will be called Lxþ and Lc; x , respectively. A similar lifting can be applied to connectivity cuts (9) on GL if the copies of all nodes of at least one cluster are contained in set W þþ yielding layered cluster cuts (11). In the following, Lc; will be x þ used to refer to model Lc; additionally containing these conx straints. 

X½δ ðWÞ Z 1

8 W D V L ⧹fu0 : u A V 1 g;

(k A C⧹f1g : fuh : u A V k ; 1 rh r Hg  W

ð11Þ

For the GMSTP, Pop [26] considered a reduced graph modeling potential inter-cluster connections to which we will refer to as cluster graph in the following. Given an instance to a GNDP with graph G ¼ ðV ; EÞ, node clusters Vi, 1 rk rK, and the associated arc set A, the corresponding cluster graph GC ¼ ðV C ; AC Þ is a directed graph containing one node for each cluster in G, i.e., V C ¼ f1; 2; …; Kg. Cluster arcs (i,j) between cluster nodes i; jA V C exist if there exists at least one arc in A connecting cluster i to cluster j, i.e., AC ¼ fði; jÞj ðu; vÞ A A; u A V i ; v A V j g, see Fig. 3 for an example. Pop [26] used a compact formulation for describing a spanning arborescence on this cluster graph together with a polynomial number of linking constraints to select nodes and arcs from the original graph to derive a valid formulation for the GMSTP. Our next model to which we will refer to as Ly is based on these ideas but uses a cutset based formulation to model the arborescence on the cluster graph. Thus, our formulation needs significantly less variables (j V j þ j Aj þ j AC j rather than j V j þ j Aj þ K  j AC j ) but comes at the price of an exponential number of (efficiently separable) constraints. Besides the variables introduced above we will also use variables yij A f0; 1g, 8 ði; jÞ A AC , that indicate whether or not a (directed) connection between two clusters is

3

Fig. 3. Cluster graph GC corresponding to the instance and the solution given in Fig. 1. Arcs not contained in the solution are not shown.

used or not. X min cuv xuv

ð1Þ

ðu;vÞ A A

s:t:

z½V i  ¼ 1

8 iA C



ð2Þ

x½δ ðuÞ ¼ zu

8 u A V⧹V 1

ð3Þ

xuv þ xvu r zu

8 u A V;

ð5Þ

X

X huv 1 Z

ðuh  1 ;vh Þ A AL ;u 2 = Vi

8 fu; vg A E X

X hvw

8 vh A V L ; 8 i A C; h Z 1; δðfvg; V i Þ a∅

ðvh ;wh þ 1 Þ A AL ;w A V i

ð7Þ 

y½δ ðiÞ ¼ 1

8 i A V C ⧹f1g



y½δ ðWÞ Z 1 x½δðV i ; V j Þ ¼ yij X

ð12Þ

8 W D V C ⧹f1g; W a ∅ 8 ði; jÞ A AC H 1 X

X huv ¼ yij

ð13Þ ð14Þ

8 ði; jÞ A AC

ð15Þ

ðu;vÞ A Aj u A V i ;j A V j h ¼ 0

ðx; y; z; XÞ A f0; 1g j Aj þ j AC j þ j V j þ j AL j

ð16Þ

Equations (12) are indegree constraints on the cluster graph ensuring that each cluster except the root cluster has exactly one incoming cluster arc. Inequalities (13) are the directed cutset constraints ensuring connectivity of the global (i.e., cluster-wise) structure of the solution and constraints (14) are the linking constraints between cluster arc and standard arc variables. To reduce the number of constraints we use constraints (15) linking layered arc variables to cluster arc variables rather than inequalities (6) which would be another (disaggregated) option to obtain a correct model. A valid formulation for the GHMSTP is obtained together with constraints (2), (3), (5) and (7) which have been discussed above for formulation Lx. Variant Lcy is obtained by þ adding layered cuts (9) while Lc; contains layered cuts (9) and y layered cluster cuts (11). We note, that while we introduce j AC j additional variables (compared to Lx), the number of cutset constraints decreases significantly and it remains to analyze the theoretical and computational consequences in Sections 4 and 5, respectively.

3. Layered cluster graph models While layered graph formulations similar to the ones introduced in the previous section have proven to be quite efficient in solving small and medium instances of different network design problem (see, e.g., [14–16,23]) their main drawback is the quite large number of variables involved making them impractical for large scale instances.

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M. Leitner / Computers & Operations Research 65 (2016) 1–18

variables yij A f0; 1g, 8 ði; jÞ A AC , with the same meaning as in model Ly. X min cuv xuv ð22Þ ðu;vÞ A A

s:t:

ð2Þ; ð3Þ; ð5Þ; ð12Þ ð14Þ H X  Y½δ ðih Þ ¼ 1 8 i A C⧹f1g

ð2Þ

h¼1

X Fig. 4. A layered cluster graph GCL corresponding to the instance given in Fig. 1 for H¼ 3. Arcs not corresponding to those contained in the exemplary solution from Fig. 1 are not shown.

Y hij  1 ZY hjw

8 ðjh ; wh þ 1 Þ A ACL ; h Z 1

ð18Þ

ðih  1 ;jh Þ A ACL ;i a w H 1 X

Y hij ¼ yij

8 ði; jÞ A AC

ð20Þ

h¼0

In order to significantly reduce the number of variables while aiming to keep the main advantage (i.e., a tight linear programming (LP) relaxation due to implicitly ensuring the hop limit) the two ILP models introduced in this section are based on the idea of a layered cluster graph in which only global (i.e., inter-cluster connections) are modeled on the layered graph. Formally, such a layered cluster graph GCL ¼ ðV CL ; ACL Þ is defined by its node set V CL ¼ f10 g [ fih j 2 r i rK; 1 rh r Hg and its arc set ACL ¼ fðih ; jh þ 1 Þ j ih ; jh þ 1 A V CL ; ði; jÞ A AC g, see Fig. 4. Our first such formulation to which we will refer to as CLx uses variables Y hij A f0; 1g, 8 ðih ; jh þ 1 Þ A ACL , 0 rh o H, which express whether or not the path from the root cluster to cluster j consists of precisely h þ 1 arcs and cluster i is the predecessor of cluster j along this path. In addition node design variables zu A f0; 1g, 8 u A V, and arc design variables xuv A f0; 1g, 8 ðu; vÞ A A, with the same meaning as in the previous models are used. X min cuv xuv ð17Þ ðu;vÞ A A

ð2Þ  ð5Þ

s:t:

H X



Y½δ ðih Þ ¼ 1

8 i A C⧹f1g

ð2Þ

Y hij ¼ x½δðV i ; V j Þ

8 ði; jÞ A AC

ð18Þ

h¼1 H 1 X h¼0

X

Y hij  1 ZY hjw

8 ðjh ; wh þ 1 Þ A ACL ; h Z1

ð19Þ

ðih  1 ;jh Þ A ACL ;i a w

ðx; z; YÞ A f0; 1g j Aj þ j V j þ j ACL j

ð20Þ

Constraints (2)–(5) ensure that an arborescence spanning exactly one node from each cluster is built, while the remaining constraints are used to model the corresponding arborescence on the layered cluster graph GL and thus guarantee that the hop constraint is met. Equations (17) ensure that for each cluster exactly one ingoing arc on VCL is selected among all layers. Equations (18) are the linking constraints between layered cluster arc and original arc variables and inequalities (19) are the aforementioned compact connectivity constraints modified to graph GCL . As for the previous models, one obtains a stronger model CLcx by additionally considering directed cutset constraints (21) on GCL. 

Y½δ ðWÞ Z1

8 W D V CL ⧹f10 g; W a ∅

ð21Þ

Along the same lines as in Section 2, two additional models to þ which we will refer to as CLxþ and CLc; x , respectively, are obtained by including cluster cuts (10). Above formulation can be adapted to simultaneously use the concept of a cluster graph and a layered cluster graph. Besides the variables of CLx, the resulting model CLy uses cluster connection

ðx; y; z; YÞ A f0; 1g j Aj þ j AC j þ j V j þ j ACL j

ð23Þ

Besides using (2), (3), (5) and (12)–(14) to ensure the existence of the arborescence spanning exactly one node from each cluster, the only difference to the previous model is given by the fact that linking constraints (22) link layered cluster arc to cluster arc variables. Similar to above, the model additionally including inequalities (21) will be called CLcy in the following. 4. Polyhedral comparison In this section, we compare the formulations proposed in Sections 2 and 3 and their variants in a theoretical sense, i.e., with respect to their LP relaxation values. Thereby, for formulation M, we denote by P(M) the polyhedron induced by its LP relaxation and by projx1 ;x2 ;…;xm ðPðMÞÞ the orthogonal projection of polyhedron P(M) onto the space ðx1 ; x2 ; …; xm Þ . The optimal value of the LP relaxation of formulation M is denoted by vðMÞ A R. Formulation M1 is at least as strong than formulation M2 if vðM 1 Þ ZvðM 2 Þ and is strictly stronger than M2 if M1 is at least as strong as M2 and there exists at least one instance for which vðM 1 Þ 4 vðM 2 Þ. M1 and M2 are incomparable if there exist instances such that vðM 1 Þ 4 vðM 2 Þ as well as instances with vðM 2 Þ 4 vðM 1 Þ. All considered variants of the different formulations are summarized in Table 1. The obtained results are summarized in Fig. 5 where a dotted edge indicates two incomparable formulations and a solid arrow that the formulation at the target is strictly stronger than the one at the source. We emphasize that besides comparing the proposed formulations our results also show that fractional solutions contained in a polyhedron corresponding to a formulations that makes use of the clustered graph GC or clustered layered graph GCL may contain paths that contain more than H arcs (considering the projection to the ðx; zÞ-space), cf. the proof of Theorems 4.3 and 4.5. The latter is not true for the formulations not using any of these two graphs. Theorems 4.1 and 4.2 summarize relations between different formulations for cases when one formulation is contained in the other. Since it is not too difficult to find exemplary solutions (to the corresponding LP relaxations) showing that the added constraints are strengthening (and some of these examples could be easily obtained by slightly modifying or reusing similar examples from the literature) we do not give explicit proofs of these two theorems. Theorem 4.1. The following relations hold: (i) (ii) (iii) (iv) (v)

Formulation Formulation Formulation Formulation Formulation

þþ þ Lc; is strictly stronger than formulation Lc; x x . c; þ c Lx is strictly stronger than formulation Lx . þ Lc; is strictly stronger than formulation Lxþ . x Lcx is strictly stronger than formulation Lx. Lxþ is strictly stronger than formulation Lx.

M. Leitner / Computers & Operations Research 65 (2016) 1–18

(vi) Formulation Lcy is strictly stronger than formulation Ly. þ (vi) Formulation Lc; is strictly stronger than formulation Lcy . y Theorem 4.2. The following relations hold: (i) (ii) (iii) (iv) (v)

Formulation Formulation Formulation Formulation Formulation

þ CLc; is strictly stronger than formulation CLcx . x þ CLc; is strictly stronger than formulation CLxþ . x CLcx is strictly stronger than formulation CLx. CLxþ is strictly stronger than formulation CLx. CLcy is strictly stronger than formulation CLy.

The next two theorems establish relations between models using cluster graph GC and there corresponding variants not using this concept. Table 1 Overview on the formulations which are considered the polyhedral comparison. Columns show whether or not cluster graph GC or layered cluster graph GCL and which types of cuts are considered.

5

Theorem 4.3. The following relations hold: (i) Formulation Lxþ is strictly stronger than formulation Ly. þ (ii) Formulation Lc; is strictly stronger than formulation Lcy . x þþ þ (iii) Formulation Lc; is strictly stronger than formulation Lc; x y . Proof. We first show that Lxþ is at least as strong as Ly, by showing that PðLxþ Þ Dprojx;X ðPðLy ÞÞ. Eq (14), i.e., yij ≔x½δðV i ; V j Þ, 8 ði; jÞ A AC , is used for each solution ðx; XÞ A PðLxþ Þ to construct a feasible assignment of the additional variables y. Thus, we need to show that constraints (12), (13) and (15) are satisfied. We note that validity of (12) and (15) is easily observed by using (14) and the fact that they correspond to (aggregated) variants of (3) and (6). Using (10), we observe that cutset constraints (13) are satisfied as well since 



y½δ ðWÞ ¼ x½δ ðfu A V j j j A WgÞZ 1:

Group

Name

GC

GCL

Node

Cluster

Lay. node

Lay. cluster

Lnx

Lx Lcx Lxþ þ Lc; x þþ Lc; x

– – – – –

– – – – –

(4) (4) (4) (4) (4)

– – (10) (10) (10)

– (9) – (9) (9)

– – – – (11)

Since the solution given in Fig. 6 is feasible for Ly but violates inequalities (10) for the cluster containing nodes u6 and u7, we conclude that Lxþ indeed is strictly stronger than Ly. Since the same argumentation as well as the example from Fig. 6 remains valid when comparing the formulations including þ connectivity cuts on the layered graph (i.e., Lc; and Lcy ), the x second statement of the theorem follows. Similarly, the third statement follows from additionally considering the example given in Fig. 7.□

Ly

þ

–

–

(13)

–

–

Theorem 4.4. The following relations hold:

Lcy

þ

–

–

(13)

(9)

–

þ Lc; y

þ

–

–

(13)

(9)

(11)

CLx

– – –

þ þ þ

(4) (4) (4)

– – (10)

– – –

– (21) –

Lny

CL nx

CLcx

CLxþ þ CLc; x CL ny

–

þ

(4)

(10)

–

(21)

CLy

þ

þ

–

(13)

–

–

CLcy

þ

þ

–

(13)

–

(21)

(i) Formulation CLxþ is strictly stronger than formulation CLy. þ (ii) Formulation CLc; is strictly stronger than formulation CLcy . x Proof. Similar as before, we first show that PðCLxþ Þ D projx;z;Y ðPðCLy ÞÞ þ c and PðCLc; x Þ D projx;z;Y ðPðCL y ÞÞ, by using the identity yij ≔x½δðV i ; V j Þ. It is immediate that (18) holds if and only if (22) holds as well. Validity of (12), (13), and (15) can be shown by the same calculations and arguments as in the proof of Theorem 4.3. To see that the formulations are indeed strictly stronger, we observe that the previously discussed solution from Fig. 6 indeed is feasible for CLy and CLcy (it is easy to derive a feasible assignment of variables Y) but violates cluster cuts (10) and thus is infeasible þ for CLxþ and CLc; x .□ Theorem 4.5. The following relations hold: (i) Formulation Lx is strictly stronger than formulation CLx. þ þþ (ii) Formulation Lc; is strictly stronger than formulation CLc; x . x

Fig. 5. Theoretical relations between the proposed formulations.

Proof. We first show that projx;z ðPðLx ÞÞ Dprojx;z ðPðCLx ÞÞ, i.e., that Lx is at least as strong as CLx. Given an arbitrary solution

Fig. 6. (a) An exemplary instance with edge costs cu1 u5 ¼ cu3 u4 ¼ M, cu3 u5 ¼ 2 M, and ce ¼1 for all other edges. (b) A feasible solution to projx ðPðLcx ÞÞ and projx ðPðLyþ ÞÞ of this instance with objective value M þ 2 for which a feasible assignment of variables X is given in (c) and a feasible assignment of variables y in (d). Solid arcs correspond to variable values of 1 and dashed arcs to variable values of 0.5. Observe that this solution is infeasible for projx ðPðLxþ ÞÞ since cluster cuts (10) are violated for W ¼ f5; 6; 7g. Each LP solution satisfying (10) has an objective value greater than 2M as it must use edge fu3 ; u5 g.

6

M. Leitner / Computers & Operations Research 65 (2016) 1–18

þ Fig. 7. (a) An exemplary instance with edge costs cu1 u5 ¼ M and ce ¼1 for all other edges. (b) A feasible solution to projx ðPðLc; y ÞÞ with objective value M þ 5=2 for H¼ 2 for which the corresponding feasible assignment of variables y is given in (c) and a feasible assignment of variables X is easy to derive. Solid arcs indicate variable values of 1 and dashed arcs variable values of 0.5. Observe that this solution is infeasible for projx ðPðLx ÞÞ since the path u1  u2  u4  u6 has length 3 and thus, no feasible assignment of variables X can exist for all models containing Lx. The optimal value of the LP relaxation of formulation Lx must exceed M since each solution must contain arc ðu1 ; u5 Þ.

Fig. 8. (a) An exemplary instance with edge costs cu1 u4 ¼ M, cu1 u3 ¼ 2M and ce ¼1 for all other edges. (b) A feasible solution to projx ðPðCLcx ÞÞ with objective value M þ 3=2 for H¼ 2 for which the corresponding feasible assignment of variables Y is given in (c). Dashed arcs indicate variable values of 0.5. Observe that this solution is infeasible for projx ðPðLx ÞÞ since compact connectivity constraints (7) are violated along the path u4  u3  u5 . The optimal value of each LP solution to this instance feasible for Lx must exceed M.

Fig. 9. (a) An exemplary instance with edge costs cu1 u5 ¼ M and ce ¼ 1 for all other edges. (b) A feasible solution to projx ðPðCLcx þ þ ÞÞ with objective value M þ 5=2 for H¼ 2 for which the corresponding feasible assignment of variables Y is given in (c). Solid arcs indicate variable values of 1 and dashed arcs variable values of 0.5. Observe that this solution is infeasible for projx ðPðLx ÞÞ since the path u1  u2  u4  u6 has length 3 and thus, no feasible assignment of variables X can exist. The optimal value of the LP relaxation of formulation Lx must exceed M since each solution must contain arc ðu1 ; u5 Þ.

ðx; z; XÞ A PðLx Þ, we will construct a solution ðx; z; YÞ A PðCLx Þ by P using Y hij ≔ ðuh ;vh þ 1 Þ A AL ;u A V i ;v A V j X huv . Observe that constraints (2)–(5) are satisfied as they are contained in both formulations. Constraints (17) hold since H X



Y½δ ðih ÞÞ ¼

h¼1

H X

ðu;vÞ A A;v A V i h ¼ 1

h¼1

¼

X



X h ½δ ðV i Þ ¼

H X

X

X

X huv ¼

xuv

ðu;vÞ A A;v A V i



x½δ ðuÞ ¼ zðV i Þ ¼ 1

u A Vi

and (18) since H 1 X

Y hij ¼

h¼0

H 1 X

X

X

X huv ¼

h ¼ 0 ðu;vÞ A A:u A V i ;v A V j

H 1 X

X

X huv ¼

ðu;vÞ A A:u A V i ;v A V j h ¼ 0

xuv :

ðu;vÞ A A:u A V i ;v A V j

¼

X

X

i A C⧹fwgðuh  1 ;vh Þ A AL ;u A V i ;v A V j

Y hjw ¼

X

X

v A V j ðvh ;kh þ 1 Þ A AL ;k A V w

X hvk r

X

X

v A V j ðuh  1 ;vh Þ A AL ;u a V w

X huv 1

Y hij  1

ðih  1 ;jh Þ A ACL ;i a w

To conclude that Lx is strictly stronger than CLx, we observe that the solution given in Fig. 8 is feasible for the LP relaxation of CLx but infeasible for Lx since it is not possible to find a valid assignment of variables X due to the path fðu4 ; u3 Þ; ðu3 ; u5 Þg where u4 and u5 are contained in the same cluster. Thus the first statement of the theorem follows. þþ Using these results, to see that Lc; is at least as strong as x c; þ CLx it only remains to show that using above transformation þþ each solution to Lc; will also satisfy layered cuts (21). The latter x holds for any set W  V CL , 10 2 = W, W a ∅, since X



Y½δ ðWÞ ¼

ðih ;jh þ 1 Þ A ACL ;j A W

Finally, the compact connectivity constraints (19) also hold for each ðjh  1 ; wh Þ A ACL , as

X

X huv 1 ¼

Y hij ¼

X

X

X huv

ðih ;jh þ 1 Þ A ACL ;j A W ðuh ;vh þ 1 Þ A AL ;u A V i ;v A V j



¼ X½δ ðfv A V j j j A WgÞ Z1: þ þþ To conclude that Lc; is strictly stronger than CLc; x , we x consider the solution given in Fig. 9 which is feasible for the LP

M. Leitner / Computers & Operations Research 65 (2016) 1–18 þ relaxation of CLc; with a value of M þ 5=2. Since this solution is x þþ infeasible for Lc; (indeed already for Lx) and the optimal value x þþ of each solution to the LP relaxation of Lc; to this instance is at x least M, the theorem follows.□

7

Theorem 4.6. The following formulations are incomparable (i) (ii) (iii) (iv)

Lcx and Lxþ CLcx and CLxþ Lcx and CLxþ Lxþ and CLcx

Proof. To see that all statements hold, we first observe that the solution given in Fig. 6 is feasible for all formulation of the statement not containing cluster cuts (10), i.e., for Lcx and CLcx , but infeasible for the Lxþ and CLxþ . Conversely, we note that connectivity cuts on layered graphs known to strengthen formulations including “standard” connectivity cuts for hop- and resource constrained spanning and Steiner tree problems, see, e.g., [14,29]. For instances with j V i j ¼ 1, i ¼ 1; …; K, or when the current LP solution is integral with respect to variables z, these results from the literature imply that there exist instances feasible for Lxþ and CLxþ but infeasible for Lcx and CLcx , see Fig. 10 for an example.□

Fig. 10. (a) A feasible solution to projx ðPðLxþ ÞÞ and projx ðPðCLxþ ÞÞ for an instance with V i ¼ fig, i A f1; 2; …; 5g and H¼3, cf [14]. A corresponding feasible assignment of variables X or Y, respectively, is given in (b). Observe that this solution is infeasible for projx ðPðLcx ÞÞ and projx ðPðCLcx ÞÞ since layered (cluster) cuts (11) and (21) are violated for W ¼ V L ⧹f10 ; 21 ; 32 g and W ¼ V CL ⧹f10 ; 21 ; 32 g, respectively.

5. Computational study Branch-and-cut algorithms (B&C) for all formulations proposed in Sections 2 and 3 have been developed in Cþ þ using IBM CPLEX 12.6. Standard settings of CPLEX have been used and each experiment has been performed on a single core within a cluster of computers, each consisting of 20 cores (2.3 GHz) and 64 GB RAM. An absolute time limit of 10,000 CPU-seconds has been applied to each experiment and CPLEX has been configured to use at most 3 GB.

Table 2 Overview on the formulations for which branch-and-cut approaches have been tested. Columns show whether or not cluster graph GC or layered cluster graph GCL and which types of cuts are considered. Group

Name

GC

GCL

Node

Cluster

Lay. node

Lay. cluster

Lnx

þ Lx;N

–

–

–

(10)

–

–

Lxþ þþ Lc; x;L

– –

– –

(4) (4)

(10) (10)

– –

– (11)

5.1. Branch-and-cut configuration

þþ Lc; x

–

–

(4)

(10)

(9)

(11)

Ly

þ

–

–

(13)

–

–

þ Lc; y;L

þ

–

–

(13)

–

(11)

þ Lc; y

þ

–

–

(13)

(9)

(11)

þ CLx;N

–

þ

–

(10)

–

–

CLxþ

–

þ

(4)

(10)

–

–

þ CLc; x

–

þ

(4)

(10)

–

(21)

CLy

þ

þ

–

(13)

–

–

CLcy

þ

þ

–

(13)

–

(21)

We initialized all models by their sets of compact constraints while the various variants of directed cutset contraints are separated by using the maximum-flow algorithm of Cherkassky and Goldberg [2]. Thereby, we add back- and nested-cuts as well as considered creep-flows to generate sparse inequalities, see [21]. To avoid tail-off effects in branch-and-cut nodes, we only separate cuts if they are violated by a value of at least 0.1 in the current LP solution. For formulation Lxc; þ þ , cuts are separated in the following order: (i) cluster cuts (10), (ii) node cuts (4), (iii) layered cluster cuts (11), and (iv) layered cuts (9). Thereby, each cut type is only considered if no violated constraints have been added for all

n

Ly

CL nx

CL ny

Table 3 Summarized results for smaller instances grouped by original instance, clustering method and hop limit, respectively. Numbers of solved instances (#solved ) and numbers of cases with best performance ð#best Þ are reported per model class. Lnx refers to the variants of Lx (i.e., those not considering the (layered) cluster graph), Lny are the variants of Ly (use the cluster graph), models CLnx make use of the layered cluster graph, and finally CL ny utilize both the cluster graph and the layered cluster graph, see Table 1. Individually best performing variant per model class is considered for each experiment. Best values are marked bold.

Instance

Clustering method

Hop limit (H)

#

#solved Lnx

Lny

CL nx

CLny

#best Lnx

Lny

CL nx

CLny

att48 eil51 st70 eil76 pr76 gr96 rat99

20 20 20 20 20 20 20

20 20 20 20 20 20 20

20 20 20 16 17 12 3

20 20 20 20 20 20 17

20 20 20 20 19 16 5

8 6 7 9 9 7 12

0 0 0 0 0 0 0

12 16 13 11 11 13 8

0 0 0 0 0 0 0

Geographical Grid, μ¼3 Grid, μ¼5 Grid, μ¼7 Grid, μ¼10

28 28 28 28 28

28 28 28 28 28

21 15 22 25 25

28 25 28 28 28

25 21 24 25 25

10 18 14 9 7

0 0 0 0 0

18 11 14 19 22

0 0 0 0 0

3 4 5 6

35 35 35 35

35 35 35 35

32 26 25 25

35 34 34 34

34 31 28 27

33 18 3 4

0 0 0 0

3 17 32 32

0 0 0 0

8

M. Leitner / Computers & Operations Research 65 (2016) 1–18

Table 4 Summarized results for larger instances grouped by original instance, clustering method and hop limit, respectively. Numbers of solved instances (#solved ) and cases with best performance ð#best Þ are reported per model class. Lnx refers to the variants of Lx (i.e., those not considering the (layered) cluster graph), Lny are the variants of Ly (use of the cluster graph), models CL nx make use of the layered cluster graph, and finally CLny utilize both the cluster graph and the layered cluster graph. Individually best performing variant per model class is considered for each experiment. Best values are marked bold.

Instance

Clustering method

# #

#solved Lnx

Lny

CLnx

CLny

#best Lnx

Lny

CLnx

CLny

kroa100 krob100 kroc100 krod100 kroe100 rd100 eil101 pr107 pr124 bier127 gr137 pr144 kroa150 krob150 pr152 u159

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20 20 19 5 16 18 18 15 12

11 13 9 7 10 11 6 15 10 13 0 7 1 0 2 0

20 20 20 20 20 20 19 20 20 19 4 20 12 12 19 10

20 17 18 18 19 20 10 20 16 16 0 12 4 1 6 0

5 10 6 6 9 10 6 0 1 2 6 0 11 9 3 7

0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0

15 10 14 14 11 10 14 11 19 16 14 19 9 11 16 13

0 0 0 0 0 0 0 5 0 2 0 1 0 0 1 0

Geographical Grid, μ¼ 3 Grid, μ¼ 5 Grid, μ¼ 7 Grid, μ¼ 10

64 64 64 64 64

52 50 56 61 64

14 2 17 38 44

53 46 52 60 64

37 30 37 44 49

14 27 24 15 11

0 0 0 1 3

50 33 37 47 49

0 4 3 1 1

3 4 5 6

80 80 80 80

76 71 69 67

45 26 23 21

65 67 70 73

60 49 45 43

53 27 8 3

1 1 1 1

20 50 70 76

6 2 1 0

Hop limit (H)

Table 5 Strength of LP relaxations on smaller instances grouped by original instance, clustering method and hop limit, respectively. Numbers of instances (#) for which LP relaxation could be solved by all variants within the time limit of 10 000 CPUseconds and average integrality gaps in percent. Integrality gaps for each considered formulation M with LP relaxation value v(M) have been computed as ðOPT  vðMÞÞ=vðMÞ. #

Integrality gap [%] Lxþ

þþ Lc; x

þ CL c; x

CLcy

Instance

att48 eil51 st70 eil76 pr76 gr96 rat99

20 20 20 20 19 19 19

2.78 8.34 7.18 6.82 4.71 4.82 9.00

2.20 7.82 5.24 5.31 1.39 1.71 2.04

3.12 8.92 7.41 7.70 5.22 5.22 8.95

20.23 22.13 26.55 32.64 28.24 27.24 30.79

Clustering method

Geographical Grid, μ ¼3 Grid, μ ¼5 Grid, μ ¼7 Grid, μ ¼10

27 27 28 28 27

7.74 5.84 6.27 5.54 5.82

5.77 1.18 3.19 3.76 4.71

8.42 4.86 6.99 6.34 6.67

30.90 12.36 22.17 30.71 37.83

3 4 5 6

35 34 34 34

7.39 6.65 5.93 4.95

3.67 3.90 3.76 3.54

8.79 6.99 5.85 4.92

25.67 27.07 27.01 27.43

Hop limit (H)

preceding variants in the current iteration. An analogous strategy is used for the other variants of model Lx by simply skipping all cut types that are not contained in the corresponding formulation. þ Similarly, for model CLc; we first separate cluster cuts (10), before x possibly considering node cuts (4) and layered cuts (21). Again, we simply skip not-contained types of cuts for other variants of CLx. Finally, in models Lcy and CLcy , layered cuts (21) are only added if no violated cuts (13) have been added in the current iteration.

Fig. 11. Distributions of CPU-times [s] for variants of Lx and CLx.

þ þþ Fig. 12. Distributions of CPU-times [s] of Lc; and CL c; by clustering method. x x;L

In case all variable values of the current LP solution are integral, the separation routine significantly simplifies due to the presence of compact connectivity constraints on corresponding layered graphs. Hence, for such solutions we only search for violated cluster cuts (10) for all variants of formulations Lx and CLx and for violated cuts (13) for the variants of Ly and CLy. Notice that we also test variants of the developed formulations considering (layered) cluster cuts but skipping (layered) node cuts. While the opposite was natural for the theoretical development and comparison of the models, these variants are more likely to perform better than variants considering (layered) node but not (layered) cluster cuts. Table 2 summarizes the formulations for which branch-and-cut approaches have been tested. Thereby, subscripts N and L indicate variants that differ from the previous

M. Leitner / Computers & Operations Research 65 (2016) 1–18

definitions by not considering node cuts or layered node cuts, respectively. In the following, we will for simplicity refer to each branch-and-cut algorithm by the abbreviation of the formulation it is defined by. 5.2. Benchmark instances We evaluate the performance of our approaches on instances from Fischetti et al. [7] originally proposed for the generalized traveling salesman problem but which have been widely used as (basis for) benchmark instances for assessing the performance of algorithmic approaches to various generalized network design problems, see, e.g., [18,19]. This benchmark set originates from Euclidean TSPlib [28] instances for which two different clustering routines (either geographical or grid clustering) have been applied. Instances with geographical clustering were created by first selecting K  ⌈j V j =5⌉ nodes at maximum distance from each other which define the seed nodes of the K clusters. All other nodes were assigned to the cluster containing its individually closest seed node. Instances based on grid clustering have been generated according to the coordinates of nodes in the plane (by dividing the plane into rectangles) and according to an input parameter μ , μ A f3; 5; 7; 10g, which determines the average number of nodes per cluster, see [5,7] for details. Thus, in total five instances have been created for each original TSPlib instance. To use these instances for the GHMSTP, the first cluster of an instance is assumed to be the root and we tested different values of H for each instance. 5.3. Initial solution Preliminary experiments showed that CPLEX fails to find feasible solutions within 10,000 s for a few, rare cases. We also observed that given an initial solution of reasonable quality, the built in general purpose heuristics are typically rather successful in finding high-quality (close to optimal) solutions. Thus, we did not implement a primal heuristic that is repeatedly called during the course of the branch-and-cut but construct an initial, feasible solution (which is passed to the solver) as follows: First a node is selected from each cluster randomly. Afterwards, a simple Prim based heuristic is applied which greedily adds the chosen node with cheapest additional costs to the partial tree if it can be reached without violating the hop-constraints. This heuristic is repeated 30 times and the overall best solution is adopted.

6. Computational results Tables 3 and 4 give a first overview on the results obtained within our computational study for smaller and larger instances, respectively. Thereby, results are grouped according to the four main classes of models discussed in the previous sections (i.e., whether or not the concept of cluster and/or layered cluster graph is used). Each line of these two tables reports how many instances could be solved by at least one of the models from each particular class (#solved ) and as well as how often a model from each class achieved the best performance (#best ). Thereby, we consider a variant as having the best performance if it could solve that particular instance to proven optimality at least as fast as any other tested variant. In case that instance could not be solved by any variant within the given time limit of 10,000 s, all variants for which the obtained optimality gap attains a minimum are considered to perform best. To facilitate a first overview on the influence of the various parameters, each of the two tables depicts

9

those results grouped according all pairs out of the three main criteria (i.e., original instance, clustering method, hop limit) within our benchmark set. From Tables 3 and 4 we conclude that all proposed model classes allow to solve most of the smaller instances to proven optimality within the given time limit of 10,000 CPU-seconds. Considering the larger instances it is easy to observe that the use of the cluster graph does not pay off computationally. As we have shown in Section 4 the corresponding variants based on Ly and CLy are dominated by other variants that do not use the cluster graph. Given the computational results, we conclude that the theoretically smaller number of connectivity cuts on the cluster graph and therefore potentially faster solution of the LP relaxations at each node of the branch-and-cut tree does not compensate the weaker bounds and larger number of nodes to consider. We also observe that the relative performance of models based on Ly and CLy improves (compared to the others) for instances with not too few nodes per cluster (i.e., grid clustering with μ A f7; 10g). The latter might indicate that models utilizing the concept of the cluster graph might pay off on instances with a large number of nodes per cluster. On the contrary, we observe that variants making use of the layered cluster graph clearly outperform the others when considering the numbers of cases in which a best performance among all variants is achieved. This trend is particularly clear when the hop limit is not too small, for larger instances and/or when considering instances with not too few nodes per cluster. Overall, though these variants do not include the theoretically strongest þþ model (i.e., Lc; ) using the layered cluster graph and therefore x reducing the number of layered connectivity cuts seems to pays off in particular for the larger and harder ones among the considered instances. We also note that from Tables 3 and 4 we cannot observe a clear trend whether the variants based on Lx or those CLx perform better with respect to the total number of instances solved in the given maximum runtime. To analyze in more detail the performance of the two best performing classes, i.e., the various variants of Lx and CL x , corresponding runtime distributions over all test instances and parameter are given in Fig. 11. We conclude that dynamically separating cutset constraints on the (clustered) layered graph þþ yields a significantly improved performance, i.e., variants L c; , x;L c; þ c; þ þ Lx , and CLx perform better than the simpler variants of the respective class. We also observe that the overall performance c; þ þ þþ of Lx;L is slightly better than the one of Lc; and x therefore conclude that considering layered node cuts (9) does þþ not pay off on the considered instances. Overall, Lc; and x;L þ CLc; yield the best performance in their respective class and x we will therefore analyze these two in more detail in the following. þ þþ Figs. 12 and 13 show runtime distributions of Lc; and CLc; x x;L for different clustering methods and all considered values of H, þ respectively. We conclude that CLc; consistently and significantly x c; þ þ outperforms Lx;L except for those instances with a very small number of nodes per cluster (i.e., grid clustering with μ ¼3) or when H ¼3. Thus, the results of our computational study clearly show that the concept of a layered cluster graph leads to

þ þþ Fig. 13. Distributions of CPU-times [s] of Lc; and CLc; by hop limit. x x;L

10

M. Leitner / Computers & Operations Research 65 (2016) 1–18

branch-and-cut algorithms with practically good performance. These findings are also confirmed by Tables 6–10 as well as Tables 11–15 (see Appendix) that detail CPU-times and potentially existing optimality gaps after termination for all tested variants and instances. To complement the theoretical study from Section 4, Table 5 provides an overview on the average integrality gaps of a subset of formulations on the smaller instances. Thereby, only instances for which the LP relaxation could be solved withing the time limit (10,000 CPU-seconds) for all reported variants are considered. In order to gain insights about the comparison of the different modeling approaches we focus on the theoretically þ þþ strongest variants of each class, i.e., formulations Lc; , CLc; x , x c þ and CLy , and we also include Lx as a baseline. Note that we þ refrain from considering Lc; since its LP relaxation could not be y solved within the time limit for a significant number of instances. From Table 5 we first observe that the theoretical dominance between the considered models is clearly reflected in the empirically observed integrality gaps. Layered node and cluster cuts þþ (9) and (11), respectively, considered in formulation Lc; but not x þ contained in Lx significantly reduce the remaining gaps in particular on larger instances and when the hop limit is relatively small. Furthermore, we conclude that using the layered cluster þ graph (CLc; x ) does not yield too weak bounds (often comparable with those of Lxþ ) while a huge deterioration is observed for CLcy which employs the cluster graph. These findings are clearly consistent with and support the previous observations regarding the superior performance of variants Lnx and CLnx within the branch-and-cut algorithms.

7. Conclusions In this paper, we studied the generalized hop-constrained minimum spanning tree problem (GHMSTP) which arises in the design of backbone telecommunication networks when the maximum path length may not exceed a predefined upper bound. Different integer programming models based on layered graph reformulations have been discussed including variants that generalize the concept of a cluster graph [26]. Several strengthening valid inequalities have been proposed and a hierarchy with respect to theoretical strength of the proposed formulations (i.e., their LP relaxation values) has been established. The results of our computational study performed on benchmark instance known from other generalized network design problems showed that the branch-and-cut approach utilizing the layered version of the cluster graph outperforms the other variants.

Acknowledgments This work has been supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office and the Austrian Science Fund (FWF) under grant I892-N23. These supports are greatly acknowledged.

Appendix See Tables 6–15.

Table 6 Detailed results for smaller instances and geographical clustering. Besides optimal solution values (Opt), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Opt

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CLxþ

þ CLc; x

CLy

CLcy

att48

3 4 5 6

12,139 11,271 10,933 10,923

0.3 0.5 1.1 2.1

0.3 0.6 1.1 2.1

0.3 0.6 1.1 2.1

0.3 0.6 1.1 2.2

1.9 3.0 2.6 2.7

14.2 1121.6 7292.6 (10.17)

13.1 302.1 15.3 60.3

0.5 0.6 0.5 0.6

0.5 0.6 0.5 0.6

0.5 0.6 0.5 0.6

1.2 2.0 1.2 1.0

1.6 1.9 1.3 2.0

eil51

3 4 5 6

142 141 139 134

5.3 11.8 14.2 11.8

6.0 15.0 15.8 12.4

7.4 14.0 25.4 12.9

7.9 15.9 31.4 14.5

17.3 55.6 48.0 47.0

6821.8 (20.41) (23.08) (19.85)

1136.8 (10.16) (23.08) (19.86)

3.4 4.0 4.1 2.7

3.8 5.0 4.5 2.7

3.7 5.3 5.6 3.5

6.4 14.1 18.7 7.5

6.8 14.6 33.4 23.3

st70

3 4 5 6

272 248 244 241

24.9 26.1 50.1 73.7

29.2 26.6 51.6 76.5

20.9 33.2 66.6 95.7

21.1 32.5 67.1 92.9

75.4 121.0 335.6 1913.7

(13.59) (25.41) (24.46) (16.44)

9995.7 (25.27) (24.45) (16.44)

27.4 28.2 22.4 19.0

33.0 32.1 27.9 22.8

53.0 19.8 30.5 23.5

23.8 35.1 60.1 37.6

27.1 27.9 54.3 77.0

eil76

3 4 5 6

213 202 198 197

146.5 120.3 500.8 480.1

173.3 125.5 664.9 506.0

94.9 325.3 340.6 577.4

100.2 364.6 354.6 597.9

2262.5 5130.0 (3.18) (7.39)

(28.43) (32.46) (26.06) (29.34)

(27.59) (32.46) (26.06) (29.34)

143.2 132.4 82.7 93.5

176.0 161.7 107.7 117.0

166.4 163.3 106.8 150.0

1159.8 2549.3 3360.0 2489.5

848.3 1743.3 (5.61)n (9.99)

pr76

3 4 5 6

53,186 49,859 47,621 47,106

12.9 121.0 48.4 87.5

13.6 137.7 47.5 78.7

12.3 112.7 52.0 95.0

12.6 113.0 51.9 96.3

230.1 2776.3 3087.3 1886.5

(18.29) (22.50) (20.81) (19.64)

(19.75) (22.52) (20.81) (19.64)

123.2 56.6 34.5 20.8

135.5 61.0 37.0 21.5

69.2 80.5 27.2 16.9

112.0 703.4 1223.0 428.0

90.9 453.4 999.1 1152.4

gr96

3 4 5 6

265 241 231 226

61.9 142.4 169.9 213.0

81.1 176.7 192.9 216.3

82.0 95.6 189.0 240.8

80.5 94.3 205.8 220.8

767.4 2464.2 9489.9 (4.83)

(15.88) (25.59) (23.63) (19.34)

(16.72) (24.85) (23.62) (19.34)

788.5 305.8 77.9 31.4

1046.5 479.4 87.5 33.2

146.1 146.9 79.4 54.2

122.4 1878.5 1225.3 1087.7

183.1 687.4 1592.5 2696.8

rat99

3 4 5 6

533 492 457 442

718.2 (5.13) (4.51) (4.73)

937.8 (5.37) (4.52) (4.39)

552.1 1323.5 1320.3 1332.6

556.1 1287.9 1209.9 1286.2

(4.97) (19.52) (19.00) (22.65)

(29.80) (40.58) (43.81) (42.19)

(29.59) (40.07) (43.86) (42.19)

1174.9 3202.7 3587.1 2224.7

1304.8 3251.9 4026.7 3901.6

762.2 1817.1 379.3 374.3

3430.2 (22.08)n (18.85)n (18.87)n

4576.9 (7.10) (7.66) (13.32)

M. Leitner / Computers & Operations Research 65 (2016) 1–18

11

Table 7 Detailed results for smaller instances and grid clustering with μ¼ 3. Besides optimal solution values (Opt), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Opt

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CLxþ

þ CLc; x

CLy

CLcy

att48

3 4 5 6

19,437 17,825 17,339 17,115

1.0 2.3 7.2 10.9

1.0 2.3 7.1 11.1

1.0 2.2 5.8 11.4

1.0 2.1 5.8 11.4

14.1 15.9 38.1 72.0

197.3 1693.9 (6.32) (9.86)

100.4 952.2 (7.09) (9.86)

2.8 2.2 4.3 3.8

3.0 2.2 4.5 4.5

3.0 2.2 5.0 5.0

6.9 6.9 17.6 22.4

4.8 7.3 16.5 29.3

eil51

3 4 5 6

278 258 251 248

5.3 4.5 8.1 5.0

7.5 4.9 11.2 6.2

5.1 5.5 8.4 8.0

6.2 5.3 9.1 8.3

30.4 33.8 29.2 19.3

359.3 1160.0 (3.59) (2.89)

174.2 913.7 4093.7 5703.9

12.0 18.2 7.2 5.0

17.8 25.4 10.3 5.9

20.7 12.3 30.3 8.6

30.4 13.9 9.0 9.6

14.2 14.4 28.4 23.1

st70

3 4 5 6

369 338 319 312

7.7 74.5 62.5 76.4

8.3 96.2 66.3 87.2

6.4 33.7 42.7 54.5

6.3 33.8 45.5 53.7

212.1 383.4 440.4 716.9

2414.1 (13.61) (10.64) (10.77)

1543.0 8814.0 (13.51) (10.79)

32.3 59.4 32.9 35.4

46.0 86.4 47.2 48.2

19.8 46.5 27.3 29.4

44.9 238.7 232.8 144.2

21.1 70.7 80.6 209.2

eil76

3 4 5 6

379 350 329 317

42.1 1159.1 274.7 52.8

57.5 1691.7 375.1 58.7

31.2 131.3 102.8 50.5

31.4 127.9 97.4 50.5

3283.9 (5.04) (0.95) 8013.6

(5.29) (12.82) (18.38) (12.44)

(6.08) (9.88) (18.38) (12.41)

587.2 4274.2 901.1 99.7

917.7 6793.8 1291.5 148.9

240.4 1148.8 271.1 138.5

1176.0 7200.3 7523.8 2348.9

493.1 (0.91) 3451.8 2276.1

pr76

3 4 5 6

72,858 66,449 62,936 61,184

44.0 1744.3 551.6 563.9

54.5 2212.1 642.0 671.7

61.4 196.4 266.3 360.6

60.5 203.4 252.0 376.7

2643.7 (7.69)n (6.40)n (5.27)

(12.81) (17.64) (15.88) (19.97)

(4.34) (14.97) (15.88) (19.88)

411.7 1907.2 937.7 1210.9

585.2 2486.6 1336.3 1661.7

183.7 231.3 180.3 343.3

743.7 (2.89) (5.97)n (1.06)

403.8 1495.8 4568.5 (2.46)

gr96

3 4 5 6

391 353 331 324

1589.7 (3.82) (3.39) (4.26)n

1700.7 (3.83) (3.48) (4.26)

440.1 1186.3 1321.9 3815.5

437.5 1199.2 1416.0 3635.9

(4.89)n (10.27)n (7.84)n (11.32)

(13.64) (15.81) (20.57) (23.92)

(13.68) (15.75) (20.69) (23.97)

4569.6 (1.79) (1.60) (3.23)n

5994.6 (2.68) (2.26) (3.99)

1504.4 652.6 970.7 2968.7

8939.3 (7.48)n (8.34)n (9.12)n

(2.45)n 8263.0 (4.66) (7.26)

rat99

3 4 5 6

791 684 639 610

(3.60) (8.97) (8.59) (7.39)

(4.23) (8.93) (8.85) (8.67)

660.2 1030.4 2115.3 1412.0

637.8 1008.7 1961.2 1611.6

(13.37)n (16.19) (18.25)n (15.01)

(24.22) (26.32) (21.67) (19.17)

(17.28) (26.36) (21.96) (19.31)

(7.11) (11.26) (9.94) (8.28)n

(8.44) (11.48) (10.96) (8.86)

9624.6 (2.86) (2.61) (2.51)

(8.83)n (13.13)n (14.92)n (13.36)

(3.57) (4.52) (9.05) (8.30)

Table 8 Detailed results for smaller instances and grid clustering with μ¼ 5. Besides optimal solution values (Opt), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Opt

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CL x;N

CL xþ

þ CL c; x

CLy

CL cy

att48

3 4 5 6

14,980 13,940 13,343 13,251

0.6 1.1 2.5 3.6

0.6 1.1 2.5 3.6

0.6 1.2 2.5 3.7

0.6 1.2 2.5 3.6

3.5 6.5 6.4 5.7

157.9 124.6 1586.2 (9.09)

33.4 120.6 1626.3 (9.09)

0.8 1.4 1.7 1.1

0.9 1.4 1.9 1.1

1.0 1.8 1.3 1.5

1.3 2.6 2.0 2.8

1.0 2.1 2.0 4.2

eil51

3 4 5 6

178 170 165 162

3.2 3.4 3.8 4.0

4.0 3.6 3.8 4.3

4.3 5.1 3.8 4.2

4.4 5.1 3.7 4.2

12.6 24.3 27.3 37.1

1647.7 (10.64) (8.21) (9.74)

578.2 2022.6 (8.21) (9.74)

3.9 3.3 2.5 1.6

4.7 4.0 2.6 1.7

3.9 4.3 3.2 2.0

8.7 9.4 5.0 6.2

5.3 11.6 6.3 10.3

st70

3 4 5 6

249 233 228 222

2.8 6.8 25.1 26.1

2.9 6.7 27.0 26.4

3.0 6.3 26.3 26.1

3.0 6.5 25.6 26.6

28.2 85.1 292.4 663.1

3579.2 (12.01) (19.13) (20.52)

2707.4 (8.30) (19.12) (20.52)

4.9 13.3 10.8 7.2

5.2 17.3 12.4 7.5

5.0 10.3 13.6 9.3

12.6 13.3 29.6 19.6

11.3 48.4 31.7 62.0

eil76

3 4 5 6

171 162 157 152

32.0 68.0 83.1 58.8

37.4 80.5 100.0 53.8

46.3 105.9 91.3 71.9

49.0 117.4 101.1 65.1

302.1 1230.0 1487.4 1794.6

(26.03) (30.47) (30.08) (23.07)

(22.86) (30.47) (30.08) (23.07)

59.1 36.9 28.2 8.6

86.5 50.1 37.9 8.5

37.1 28.3 26.3 14.2

177.9 299.5 296.8 144.8

199.4 400.3 949.7 314.6

pr76

3 4 5 6

37,376 34,399 32,204 31,286

23.9 128.3 108.3 109.9

27.0 136.7 114.9 104.0

16.3 88.0 66.9 94.4

16.3 88.4 69.0 92.5

200.8 6282.6 1777.4 4409.4

(21.96) (30.50) (29.16) (27.11)

(20.16) (26.34) (29.06) (27.11)

59.9 128.7 28.8 25.3

81.2 174.0 35.6 29.4

78.0 95.8 35.2 19.8

86.3 678.3 344.2 133.2

107.0 897.1 964.3 2321.2

gr96

3 4 5 6

283 264 253 249

189.7 2091.8 1194.4 2381.4

226.3 2396.7 1488.3 2818.1

99.8 228.1 588.5 1599.4

87.6 218.8 595.8 1485.0

2922.6 (8.37)n (10.43) (9.73)

(16.04) (20.15) (35.98) (29.83)

(15.87) (20.08) (35.92) (29.83)

463.1 983.6 492.0 1359.0

593.5 1306.1 615.3 1791.4

252.4 817.6 867.4 523.1

2315.1 (7.78)n (6.28)n (9.17)n

827.5 4691.2 (3.58) (7.30)

rat99

3 4 5 6

564 489 466 454

6309.9 3481.0 6767.3 (1.56)

6577.2 6071.4 9325.9 (1.26)

474.8 464.9 653.9 756.5

494.0 446.7 660.2 834.7

5538.6 (13.31) (14.31) (15.56)

(24.30) (25.02) (33.99) (29.84)

(25.36) (25.25) (34.11) (29.84)

(5.15) 3759.5 7729.4 8026.8

(4.25) 4864.1 9822.5 (2.31)

5019.1 1360.6 2105.6 5032.0

(9.89)n (9.13)n (9.60)n (12.00)n

4448.0 4626.6 (6.30) (11.76)

12

M. Leitner / Computers & Operations Research 65 (2016) 1–18

Table 9 Detailed results for smaller instances and grid clustering with μ¼7. Besides optimal solution values (Opt), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Opt

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CLxþ

þ CLc; x

CLy

CLcy

att48

3 4 5 6

7082 6867 6667 6667

0.2 0.5 0.5 0.8

0.2 0.4 0.5 0.9

0.2 0.4 0.5 0.8

0.2 0.4 0.5 0.8

1.5 2.2 2.4 2.0

311.4 41.7 184.5 9.7

3.6 45.9 198.3 21.3

0.3 0.3 0.3 0.3

0.3 0.3 0.3 0.3

0.3 0.3 0.3 0.3

0.6 0.7 0.7 0.9

0.7 1.0 0.8 1.0

eil51

3 4 5 6

100 100 100 100

0.6 1.6 1.9 4.3

0.7 1.5 2.0 4.5

0.7 1.5 1.9 4.4

0.6 1.6 2.0 4.5

4.0 10.2 13.6 11.9

2287.1 (13.32) (14.33) (9.89)

212.2 8632.6 (7.92) (12.32)

0.7 0.9 0.8 1.0

0.7 0.8 0.8 1.1

0.7 1.2 0.8 1.0

1.6 2.3 2.4 2.4

1.6 3.8 3.6 5.2

st70

3 4 5 6

249 233 228 222

2.8 6.8 25.1 27.3

2.9 7.0 26.0 26.4

3.0 6.3 25.7 27.5

3.0 6.5 26.0 27.1

28.9 87.0 277.0 652.5

3529.1 (12.01) (19.22) (20.52)

2819.6 (8.15) (19.01) (20.52)

5.0 13.2 10.6 7.2

5.2 17.1 12.4 7.4

5.1 10.2 13.2 9.3

12.5 13.1 29.9 19.6

11.4 48.1 31.8 61.9

eil76

3 4 5 6

171 162 157 152

33.0 69.0 84.8 61.4

37.9 83.9 100.8 61.7

46.8 107.0 90.4 74.3

48.4 116.6 101.5 66.9

302.0 1240.4 1468.1 1782.1

(26.03) (30.47) (30.08) (23.07)

(20.82) (30.47) (30.08) (23.07)

59.3 37.9 28.1 8.5

87.3 50.4 37.6 8.5

37.2 28.3 26.2 14.4

185.4 293.6 291.4 143.0

207.5 400.1 955.8 311.1

pr76

3 4 5 6

37,376 34,399 32,204 31,286

23.9 128.9 107.9 107.4

27.2 136.0 114.7 104.2

16.3 87.4 67.7 100.0

16.2 87.4 63.5 92.3

200.2 6463.1 2023.7 4882.1

(21.96) (30.50) (29.05) (27.11)

(20.11) (26.34) (29.16) (27.11)

59.9 129.2 28.6 25.4

80.9 175.0 35.6 29.4

77.7 96.0 35.3 19.8

84.2 736.3 347.4 137.1

107.7 902.6 967.5 2260.4

gr96

3 4 5 6

204 192 190 187

34.3 98.8 111.6 147.0

35.8 99.4 118.8 144.1

42.0 121.4 330.2 148.4

42.3 127.7 327.5 140.4

938.6 1420.8 6090.5 3740.9

(24.95) (31.33) (30.32) (25.26)

(21.23) (31.33) (30.32) (25.26)

76.7 46.1 28.2 23.5

89.0 52.4 29.9 23.8

83.9 54.9 29.2 26.8

277.9 476.0 725.9 776.3

211.5 575.5 2472.0 (1.87)

rat99

3 4 5 6

395 361 340 328

572.8 2148.7 1373.8 726.3

711.5 2190.3 1526.3 876.8

203.6 947.5 544.6 526.3

206.6 1028.0 614.2 519.9

1251.1 (14.45)n (18.21) (14.00)

(31.56) (41.38) (38.29) (37.00)

(36.25) (41.23) (38.41) (37.00)

593.6 2115.3 671.2 224.5

808.4 2763.1 1098.8 296.2

471.0 706.0 495.1 204.4

1855.1 (15.23)n (11.52)n (16.45)n

1041.1 (5.99)n (15.11)n (11.31)n

Table 10 Detailed results for smaller instances and grid clustering with μ¼10. Besides optimal solution values (Opt), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Opt

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CLxþ

þ CL c; x

CLy

CLcy

att48

3 4 5 6

7082 6867 6667 6667

0.2 0.4 0.5 0.8

0.2 0.4 0.5 0.8

0.2 0.4 0.5 0.8

0.2 0.4 0.5 0.8

1.5 2.1 2.3 2.1

274.6 41.3 188.9 9.9

3.8 47.7 209.0 21.6

0.3 0.3 0.3 0.3

0.3 0.3 0.3 0.3

0.3 0.3 0.3 0.3

0.7 0.7 0.7 0.8

0.7 0.9 0.8 1.0

eil51

3 4 5 6

100 100 100 100

0.7 1.5 2.0 4.5

0.7 1.5 1.8 4.4

0.7 1.6 1.9 4.3

0.7 1.6 2.0 4.5

4.0 10.6 14.9 11.8

2030.8 (13.32) (14.33) (9.89)

212.3 7061.3 (7.92) (11.29)

0.7 0.8 0.8 1.0

0.7 0.9 0.8 1.1

0.7 1.3 0.8 1.1

1.6 2.3 2.4 2.4

1.5 3.8 3.7 5.1

st70

3 4 5 6

149 148 147 147

2.3 3.7 8.3 16.3

2.3 3.8 8.4 16.4

3.0 3.8 8.9 17.1

3.0 3.8 9.0 17.1

19.8 31.5 66.0 148.6

(34.34) (23.70) (28.33) (26.30)

2999.0 (23.70) (28.33) (21.96)

1.9 3.0 1.6 2.6

2.0 3.1 1.5 2.6

1.9 3.3 1.8 2.6

4.7 5.5 11.2 6.3

3.7 7.1 10.6 12.9

eil76

3 4 5 6

101 95 94 94

3.5 4.2 14.3 22.8

3.6 4.0 14.2 22.6

3.2 4.0 14.1 23.0

3.2 4.0 14.2 22.9

49.4 176.6 305.7 319.5

(32.15) (30.12) (31.73) (33.43)

(30.36) (30.12) (31.89) (35.90)

4.8 2.8 3.2 2.7

5.1 2.8 3.2 2.7

5.6 2.8 3.3 2.7

17.4 21.5 30.1 36.6

16.1 39.0 64.2 95.2

pr76

3 4 5 6

23,395 22,111 21,068 20,743

4.6 10.1 16.1 26.8

4.8 10.6 15.7 26.6

4.0 8.3 16.1 26.2

4.0 8.4 15.7 25.4

45.1 161.0 449.5 371.4

(22.90) (29.05) (30.36) (26.65)

(15.06) (27.70) (30.36) (26.66)

8.8 8.6 4.2 4.6

9.3 8.8 4.3 4.6

10.2 9.4 4.2 4.6

16.5 31.2 20.0 17.2

11.9 24.7 21.4 34.6

gr96

3 4 5 6

204 192 190 187

33.6 92.4 123.2 146.1

36.5 97.9 120.7 138.4

41.3 118.5 351.2 147.0

41.8 120.6 345.8 154.1

942.2 1452.2 6028.2 4049.5

(24.95) (31.33) (30.32) (25.26)

(20.85) (31.33) (30.32) (25.26)

75.2 46.8 27.3 23.7

89.3 54.5 29.7 24.2

83.4 53.7 29.2 26.7

269.9 462.5 738.2 759.8

221.3 570.4 2333.6 (2.72)

rat99

3 4 5 6

395 361 340 328

588.3 2043.6 1584.8 737.9

753.2 2303.8 1296.3 887.9

204.4 992.9 608.4 550.6

207.4 931.4 583.2 570.3

1343.3 (14.45)n (18.52) (14.71)

(32.99) (41.41) (38.42) (37.00)

(30.36) (41.15) (38.43) (37.00)

582.5 1903.0 677.6 224.1

837.7 2901.2 1094.7 301.5

476.5 701.5 510.6 205.9

1832.7 (15.23)n (11.52)n (16.45)n

1080.8 (5.99)n (15.11)n (11.78)

M. Leitner / Computers & Operations Research 65 (2016) 1–18

13

Table 11 Detailed results for larger instances and geographical clustering. Besides best solution values (Best), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Best

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CLxþ

þ CLc; x

CLy

CLcy

kroa100

3 4 5 6

9016 8525 8229 8047

167.0 2014.9 392.1 319.3

190.3 2126.0 396.0 309.9

99.2 411.5 588.5 490.5

109.7 438.4 648.6 758.3

1036.7 (11.83)n (12.97)n (5.96)

(24.52) (28.03) (29.05) (27.80)

(24.45) (28.03) (29.05) (27.80)

87.0 196.0 88.3 69.4

108.6 279.8 110.2 79.2

133.1 163.5 139.4 106.3

350.7 3711.1 5647.5 2598.0

882.5 3716.6 (4.95)n (2.79)

krob100

3 4 5 6

9937 9234 8548 8339

569.5 (1.02) 1044.5 496.3

771.5 (1.26) 1066.3 699.8

234.9 1108.3 809.5 703.1

245.6 1190.1 767.1 766.9

1941.1 (13.70)n (8.74) (6.09)

(30.11) (35.77) (32.11) (27.09)

(24.17) (35.79) (32.11) (27.09)

281.4 955.0 182.1 73.4

379.2 1249.8 253.0 87.1

580.9 602.7 106.3 76.8

1404.7 (7.76)n 2524.5 2338.3

1407.2 6380.2 3047.1 (1.10)

kroc100

3 4 5 6

9389 8417 8126 8041

48.9 55.3 61.5 113.7

60.6 50.1 61.9 110.2

42.8 42.3 63.0 109.0

42.1 53.5 63.1 112.1

3357.2 7425.8 (3.01) (7.70)

(20.33) (30.50) (31.29) (31.94)

(24.02) (30.51) (31.29) (31.94)

106.6 23.7 15.5 15.1

136.5 25.9 15.4 14.7

106.5 21.4 15.4 14.7

(0.88) 4004.8 1772.8 3271.3

2331.7 2330.7 5405.5 6845.5

krod100

3 4 5 6

9339 8393 8075 7868

355.7 444.4 424.9 241.7

405.9 488.3 488.1 242.3

117.0 253.6 461.1 344.9

115.1 249.1 480.7 367.4

(8.83)n (8.56) (9.61)n (7.61)

(18.62) (29.96) (27.83) (22.19)

(20.41) (29.89) (27.83) (22.19)

341.7 168.8 178.8 81.3

404.8 213.6 215.7 97.2

172.2 131.4 226.4 101.9

826.5 (4.49)n (7.26)n 3941.6

1045.9 1968.9 3980.8 4512.2

kroe100

3 4 5 6

10194 9457 9008 8814

376.1 8859.0 5087.1 6661.4

486.4 (0.87) 7401.8 7264.6

100.7 619.8 719.0 2382.8

96.7 652.8 761.1 2043.2

1165.9 (8.09)n (13.11) (11.09)n

(18.24) (23.14) (30.31) (31.76)

(21.00) (23.91) (30.31) (31.76)

349.7 1161.0 834.0 613.7

453.1 1453.4 1147.7 889.1

204.1 302.2 396.0 443.4

825.0 (6.95)n (13.14)n (10.38)n

370.5 4515.9 8675.5 (7.97)

rd100

3 4 5 6

3321 3033 2952 2857

39.7 96.3 202.3 157.2

43.6 103.2 206.4 177.8

41.4 97.2 172.5 185.1

41.2 95.2 178.7 175.1

1199.7 (8.13)n (8.18) (12.64)

(19.00) (31.17) (25.54) (26.91)

(17.90) (31.17) (25.53) (26.91)

92.6 178.0 160.9 46.8

123.3 241.9 204.1 49.4

119.4 76.0 100.1 56.1

552.6 (3.98)n (8.44)n 1171.0

367.6 717.1 7367.0 5341.0

eil101

3 4 5 6

234 221 214 212

195.1 545.4 660.5 1085.7

229.0 656.5 794.8 1407.7

270.3 838.9 789.4 1866.6

279.4 950.1 845.1 2284.4

(8.69)n (16.30)n (10.55) (10.06)

(21.49) (27.48) (30.19) (30.28)

(34.67) (27.48) (30.19) (30.28)

407.0 515.3 178.8 111.1

485.3 606.2 215.2 138.7

504.3 384.1 199.1 101.2

2844.4 (15.97)n (8.93)n (7.12)n

3594.9 (10.10)n (9.77) (13.23)

pr107

3 4 5 6

27,574 24,818 22,970 21,723

(6.83) (8.15) (4.44) 7670.2

(7.53) (8.35) (4.14) 8719.0

980.9 2837.8 3716.8 1881.6

921.8 2846.8 3476.6 1642.4

496.9 5048.7 (5.56) 2846.9

(5.55) (13.96) (23.95) (11.47)

(4.18) (15.93) (23.95) (11.47)

613.1 5157.3 788.0 382.2

736.6 6719.8 969.7 472.4

203.4 313.1 257.0 237.2

490.7 (4.85)n (4.79)n (3.17)n

263.1 1002.1 5304.3 3173.1

pr124

3 4 5 6

40,583 36,115 34,057 32,692

(3.10) (5.96) (6.94) (4.68)

(3.60) (6.47) (7.21) (4.81)

2345.2 3399.3 4822.5 3600.7

2192.3 3349.8 5186.9 3715.6

8714.8 (13.44)n (14.84) (12.86)

(12.59) (27.80) (28.54) (22.02)

(16.91) (27.99) (28.54) (22.02)

4154.1 9878.3 (2.88) (2.55)

5770.2 (2.40) (3.37) (3.10)

539.3 1255.4 1869.3 1600.7

(6.68)n (13.68)n (13.95)n (14.06)n

3503.5 (4.25)n (7.95)n (10.05)n

bier127

3 4 5 6

65,133 60,228 58,905 58,343

5726.4 1832.8 4601.8 9360.0

9213.2 3143.2 6133.1 (1.43)

(2.46) 2200.5 5998.8 (1.54)

(2.89) 2727.4 7749.5 (2.03)

604.7 961.3 2942.4 (1.98)n

(4.73) (6.01) (7.06) (8.62)

8314.0 (5.97) (6.78) (8.62)

233.9 94.7 79.5 63.3

323.1 124.7 100.2 74.1

239.4 79.3 107.9 79.2

480.4 292.4 452.1 1064.5

457.2 269.2 732.6 1022.3

gr137

3 4 5 6

504 451 416 395

(15.05) (21.53) (21.08) (17.89)

(14.89) (21.53) (21.08) (17.89)

(3.51) (15.43) (17.15) (12.21)

(3.05) (15.86) (16.93) (11.59)

(18.97) (33.50) (35.04) (36.81)

(37.45) (39.24) (38.11) (39.60)

(37.49) (39.79) (38.13) (39.61)

(12.47) (17.29) (20.40) (15.39)

(12.36) (18.44) (20.40) (16.01)

(9.21) (9.56) (6.16) (4.67)

(21.59)n (29.80)n (31.26)n (32.41)n

(11.07)n (16.62)n (14.97) (18.20)

pr144

3 4 5 6

51,644 46,594 43,697 42,634

(4.56) (7.31) (4.30) (3.96)

(4.58) (6.91) (4.30) (3.71)

3162.6 (2.09) 6510.2 8559.2

3095.0 (2.03) 7109.6 8818.2

3296.3 (6.82) (9.80) (6.43)

(7.75) (21.50) (18.61) (16.08)

(8.38) (21.50) (18.61) (16.08)

464.1 7162.0 (1.09) (1.22)

623.5 8715.5 (0.99) (1.69)

350.4 1554.5 798.7 1029.4

6182.4 (5.35)n (6.13)n (8.30)n

1518.5 7594.3 (1.66)n (2.29)n

kroa150

3 4 5 6

12,150 11,170 10,699 10,290

(4.53) (4.46) (3.55) (1.95)

(4.76) (4.72) (3.55) (3.53)

3192.0 8311.0 (1.40) (2.43)

2730.2 6850.9 (1.40) (2.43)

(17.17)n (24.75) (24.93) (33.86)

(41.46) (42.63) (37.39) (37.18)

(41.46) (42.63) (37.39) (37.18)

(2.66) (5.51) (3.05) 5756.6

(4.51) (5.96) (3.54) 9075.9

8019.0 (3.54) (2.51) 6036.5

(15.54)n (20.14)n (21.50)n (22.43)n

(13.56)n (15.67)n (17.14)n (18.40)

krob150

3 4 5 6

13,022 11,760 11,160 10,765

(2.59) (6.59) (6.63) (8.39)

(3.73) (6.59) (6.07) (7.48)

2193.9 5943.9 8989.9 (1.60)

2373.3 5534.4 8695.9 (1.58)

(18.57)n (28.88) (28.07) (28.09)

(42.05) (44.95) (41.96) (33.77)

(42.05) (44.95) (41.96) (33.77)

(6.77) (5.67) (6.79) (4.16)

(8.41) (6.43) (6.15) (3.65)

(4.41) (3.83) 7459.7 4480.3

(21.80)n (24.08)n (19.94)n (24.77)n

(15.29)n (17.87)n (20.15)n (20.97)

pr152

3 4 5 6

53,825 46,053 44,033 42,375

(7.63) (5.81) (10.79) (11.41)

(8.30) (6.01) (10.79) (11.41)

4124.4 8043.4 (2.27) (4.84)

4855.7 7599.8 (0.23) (4.02)

(7.58)n (9.50) (14.29) (13.32)

(11.17) (30.17) (40.97) (25.56)

(10.80) (30.17) (40.97) (25.56)

(7.04) (3.34) (6.88) (3.61)

(7.54) (3.84) (6.80) (4.25)

1613.5 1488.2 2953.6 6072.1

(5.37)n (7.92)n (13.49)n (8.85)

1973.0 2608.4 (4.54)n (3.66)

u159

3 4 5 6

26,545 23,144 21,829 20,447

(8.10) (11.34) (11.53) (5.84)

(8.38) (11.34) (10.36) (5.27)

6468.1 8391.1 (5.28) (11.91)

7370.6 9253.5 (5.24) (12.88)

(21.20)n (27.01) (28.19) (37.28)

(36.81) (42.56) (50.32) (48.63)

(36.81) (42.56) (50.32) (48.63)

(13.22) (10.49) (10.38) (5.27)

(13.78) (9.15) (10.07) (6.03)

(9.28) (6.77) (4.45) 7060.2

(17.42)n (22.89)n (25.59)n (22.94)n

(12.38)n (11.63)n (16.65) (18.22)

14

M. Leitner / Computers & Operations Research 65 (2016) 1–18

Table 12 Detailed results for larger instances and grid clustering with μ¼ 3. Besides best solution values (Best), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Best

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CLxþ

þ CLc; x

CLy

CL cy

kroa100

3 4 5 6

15,651 13,831 13,133 12,736

241.9 419.3 1315.0 522.0

386.8 650.0 1960.5 562.5

72.3 142.8 308.7 381.0

71.7 147.4 295.2 385.5

3990.7 (0.66) (4.03)n (2.01)

(2.44) (5.30) (17.11) (13.67)

6757.3 (6.17) (17.24) (13.57)

197.1 1541.3 821.3 823.5

309.6 2475.6 1289.3 1227.7

165.2 201.1 257.6 211.3

529.9 3089.7 5693.5 4667.6

173.2 342.5 1289.6 2511.7

krob100

3 4 5 6

17,200 15,198 14,438 13,880

3911.3 (2.55) (4.08) (2.31)

6533.9 (3.07) (4.42) (2.81)

305.1 292.6 1436.7 1538.4

309.2 285.1 1348.0 1463.0

(8.35)n (10.05)n (11.84)n (10.16)n

(8.81) (9.91) (16.51) (18.55)

(6.94) (18.80) (16.44) (18.50)

(1.15) (2.40) (5.07) (3.27)

(2.64) (3.51) (5.56) (4.06)

1431.4 1049.5 3524.2 1835.7

(3.80)n (7.63)n (10.55)n (9.01)n

5745.5 (0.23) (2.60) (4.32)

kroc100

3 4 5 6

16,775 14,916 13,911 13,362

4792.9 (2.81) (2.27) (1.34)

6498.6 (2.99) (2.59) (1.59)

187.2 557.2 967.8 1015.6

205.0 600.2 1013.3 1032.2

(3.61)n (6.73)n (5.87) (3.64)n

(6.66) (5.86) (12.10) (5.45)

(5.55) (7.12) (18.45) (5.23)

2348.4 (1.26) (1.86) 3531.5

3958.8 (2.71) (2.81) 5184.1

374.9 335.2 307.2 414.9

5165.0 (3.66) (3.86)n (2.55)n

282.2 1273.9 401.3 1249.6

krod100

3 4 5 6

16,809 14,861 13,874 13,284

1564.0 (3.83) (4.41) (3.88)n

2141.4 (4.01) (4.60) (4.20)

197.2 965.8 1206.9 1727.3

201.0 1072.6 1437.3 1639.2

(3.27) (10.26)n (8.65) (7.75)n

(6.08) (10.20) (10.42) (9.80)

(4.09) (9.10) (10.39) (9.19)

3129.7 (4.21) (4.28) (2.90)

4562.5 (5.51) (5.54) (3.69)

579.3 809.6 550.7 609.4

4013.2 (6.77)n (7.70)n (6.06)n

649.6 3271.3 3647.2 (2.92)

kroe100

3 4 5 6

17,201 15,212 14,225 13,681

3053.4 (4.03) (4.22) (3.98)

4923.2 (4.65) (4.52) (4.19)

93.9 423.5 957.4 1338.1

93.7 390.7 1074.0 1392.5

(5.08)n (10.09)n (9.89)n (8.99)n

(6.57) (18.56) (15.74) (14.08)

(6.06) (18.46) (15.62) (14.29)

2653.8 8725.5 (3.51) (2.85)

4391.1 (1.75) (4.83) (3.58)

560.5 792.1 865.0 938.2

4828.8 (6.35)n (7.24)n (7.45)n

724.8 6176.4 8388.8 7452.0

rd100

3 4 5 6

5564 4917 4592 4371

1073.9 (1.30) 9026.3 1408.6

1795.0 (2.18) (0.79) 1974.6

158.6 348.2 801.9 273.9

156.0 347.1 691.5 278.6

(6.00)n (9.10)n (8.87) (5.42)

(10.65) (21.44) (18.42) (18.51)

(9.28) (21.44) (18.26) (18.09)

2229.2 6891.3 (0.61) 1630.3

3562.4 (1.59) (2.21) 2655.1

264.6 443.0 437.3 318.7

6180.8 (4.80)n (7.70)n (2.40)

2892.7 3006.1 4768.4 3056.5

eil101

3 4 5 6

351 324 311 306

693.5 1934.8 1851.8 1617.1

785.3 2728.8 1189.4 2194.2

446.7 1073.3 1426.5 2156.7

479.9 1089.5 1572.0 2737.2

(5.95) (9.32) (10.07) (8.83)

(15.35) (22.77) (17.47) (19.43)

(13.73) (22.76) (17.55) (19.43)

(0.95) (1.80) 2607.9 1226.2

(2.09) (2.52) 3640.8 1927.6

(1.84) 4820.7 3178.3 1457.1

(4.03) (9.95)n (7.74)n (7.68)n

(5.62)n (6.66) (5.67) (8.40)

pr107

3 4 5 6

36,037 31,195 29,121 27,501

(8.19) (16.10)n (15.50) (13.14)

(9.04) (15.69) (15.67) (12.65)

879.3 2383.1 4149.4 5758.0

873.5 2250.9 3003.2 5843.3

(8.82)n (14.72)n (17.52)n (13.93)n

6074.8 (4.06) (13.36) (12.04)

5249.3 (1.67) (13.37) (12.83)

9333.9 (12.05) (15.79)n (11.62)n

(2.54) (13.13) (16.40) (11.97)

165.5 367.0 515.3 663.2

1715.6 (10.81)n (16.97)n (12.46)n

51.1 143.6 487.8 747.5

pr124

3 4 5 6

52,279 46,911 44,119 41,999

5045.9 (7.65) (6.99) (5.25)

7111.2 (7.91) (6.92) (5.41)

595.5 1500.8 2186.7 3776.0

610.6 1314.9 2266.7 3883.5

(3.97)n (12.16)n (10.13) (9.01)

(0.82) (21.91) (12.29) (17.53)

(0.77) (21.88) (13.08) (17.48)

2953.8 (5.70) (5.74) (4.57)

4835.7 (6.70) (6.64) (4.93)

211.2 654.6 758.8 655.5

6838.5 (8.69)n (10.55)n (9.49)n

385.7 2495.2 9294.2 (1.87)

bier127

3 4 5 6

89,013 81,024 76,793 74,595

1459.6 (1.67) (2.11) (1.79)

2048.4 (2.23) (2.24) (2.30)

2818.1 3422.3 2939.9 2688.5

3454.6 4101.0 3684.6 3540.2

(6.17)n (7.39)n (6.04)n (5.70)

(8.96) (10.51) (13.02) (14.61)

(7.30) (11.29) (13.01) (14.63)

(1.09) (2.00) (0.65) (0.75)

(1.66) (2.49) (1.24) (1.15)

(0.68) 6267.0 1989.3 2187.0

(4.77)n (4.91)n (5.12)n (4.57)n

(3.06)n (4.30)n (3.12)n (2.70)

gr137

3 4 5 6

651 572 529 503

(12.66) (23.08) (26.58) (22.64)

(12.73) (23.08) (26.58) (22.64)

(1.39) (11.82) (20.26) (22.42)

(1.21) (12.65) (19.60) (23.19)

(25.51)n (36.21) (32.16) (35.33)

(41.06) (51.34) (44.20) (40.53)

(41.03) (52.00) (43.65) (40.60)

(20.24) (23.19) (20.34) (18.46)

(21.22) (23.94) (20.34) (19.91)

(9.23) (10.55) (9.89) (8.13)

(19.38) (31.21)n (30.16)n (32.66)n

(10.53) (14.46) (17.15) (19.24)

pr144

3 4 5 6

61,456 53,301 49,232 47,284

(7.03) (9.82) (11.13) (5.84)

(7.29) (9.70) (9.89) (6.11)

3534.4 (1.45) (10.71) (13.50)

3621.3 (2.50) (10.56) (13.78)

(7.65)n (13.27)n (13.36)n (11.95)

(4.31) (19.92) (20.34) (22.07)

(3.53) (20.19) (20.98) (22.07)

(1.76) (6.49) (6.91) (3.20)

(2.67) (7.20) (7.03) (3.88)

1612.4 4169.8 1472.1 1663.1

(4.63)n (10.71)n (8.78)n (8.03)n

3059.1 (0.45) (1.01) (2.48)

kroa150

3 4 5 6

19,073 17,041 15,913 15,270

(4.92) (7.77) (6.04) (5.16)

(5.37) (7.84) (6.17) (5.88)

2681.1 6887.3 6645.5 6695.6

2721.9 7045.5 6643.8 7714.3

(13.35) (18.09) (15.39) (17.49)

(19.39) (31.35) (26.54) (25.23)

(19.47) (31.44) (26.44) (25.23)

(8.87) (10.23) (7.19) (5.45)

(10.27) (10.36) (7.60) (5.69)

(5.15) (4.65) (1.25) (1.13)

(12.21)n (15.72)n (13.39)n (13.78)n

(6.99) (9.31) (8.44) (10.86)

krob150

3 4 5 6

20,310 17,605 16,326 15,728

(7.41) (10.80) (7.55) (7.67)

(7.61) (11.96) (8.84) (7.67)

5073.2 7926.1 8857.0 (25.60)

4220.3 7084.2 9116.1 (25.44)

(17.42)n (18.75) (19.76) (22.13)

(25.62) (30.56) (31.24) (29.42)

(25.70) (29.90) (31.28) (29.42)

(10.60) (10.87) (8.48) (8.01)

(12.17) (13.51) (9.04) (8.71)

(6.88) (5.22) (1.43) (4.35)

(12.91) (18.28)n (16.47)n (12.31)

(7.43) (7.47) (9.83) (10.49)

pr152

3 4 5 6

73,072 58,836 53,112 50,502

(16.05) (20.39) (15.67) (14.18)

(16.05) (20.43) (15.99) (14.18)

5781.3 (17.71) (25.21) (19.18)

6741.2 (17.12) (25.23) (19.89)

(14.60)n (21.90)n (15.38)n (13.72)

(14.64) (26.43) (29.23) (29.85)

(14.03) (26.71) (29.37) (30.23)

(10.94) (12.33) (11.55) (8.14)

(11.63) (12.37) (11.20) (8.47)

2943.8 4761.2 3080.7 5050.5

(8.90) (11.84) (13.33)n (10.67)n

1020.1 7457.9 (1.11) (0.64)

u159

3 4 5 6

37,096 31,808 29,863 28,860

(9.75) (16.40) (19.23) (20.84)

(10.57) (16.40) (17.33) (20.84)

5798.5 (8.94) (13.92) (17.68)

5456.4 (10.91) (14.27) (16.90)

(18.88)n (26.11) (29.73) (33.22)

(29.71) (31.00) (45.66) (33.78)

(29.91) (30.71) (45.79) (33.72)

(12.50) (15.73) (18.00) (13.74)

(12.73) (17.63) (17.68) (14.44)

(4.72) (2.40) (3.97) (4.53)

(16.83)n (23.04)n (24.81)n (23.89)n

(5.45) (10.71) (14.62) (17.65)

M. Leitner / Computers & Operations Research 65 (2016) 1–18

15

Table 13 Detailed results for larger instances and grid clustering with μ¼ 5. Besides best solution values (Best), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Best

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CL x;N

CLxþ

þ CL c; x

CLy

CLcy

kroa100

3 4 5 6

9379 8798 8504 8333

58.7 404.8 422.0 498.4

67.1 465.8 760.3 552.6

60.8 177.8 251.4 809.0

63.8 179.9 258.5 641.5

685.6 (5.48)n (6.67) (4.30)

(13.26) (19.22) (25.32) (27.47)

(13.12) (27.18) (25.30) (27.47)

77.0 260.1 218.8 164.7

84.6 347.0 290.5 234.2

94.0 120.1 200.3 367.6

87.3 1083.9 3322.0 2604.6

67.1 567.6 1844.8 (2.41)

krob100

3 4 5 6

9837 8943 8603 8265

277.3 501.8 1233.6 256.0

447.9 702.9 1512.9 262.5

118.3 345.4 396.3 333.4

126.3 384.0 334.8 382.6

492.0 2961.7 8307.2 3671.3

(19.15) (20.27) (19.49) (20.36)

(12.58) (20.29) (19.49) (20.37)

188.3 307.9 628.8 102.0

302.5 463.0 981.4 138.3

132.2 219.7 211.3 157.4

129.4 423.1 1214.9 219.1

198.2 785.7 882.1 1045.0

kroc100

3 4 5 6

10,183 8943 8534 8295

172.6 255.8 386.8 335.2

225.3 297.4 459.1 328.6

132.0 122.2 305.1 345.5

135.3 125.0 285.4 344.9

4672.7 (9.05)n (9.28) (11.84)

(22.65) (43.09) (40.67) (32.91)

(23.57) (43.59) (40.67) (32.91)

367.6 284.3 156.0 89.3

554.8 367.0 211.9 109.4

357.9 178.9 177.4 132.3

4920.1 (1.14) (6.27)n (6.07)n

2771.5 9636.2 (8.51) (6.17)

krod100

3 4 5 6

10,212 9506 9165 8998

18.1 236.0 285.9 758.5

19.8 297.2 341.9 882.1

20.0 129.9 289.5 588.2

19.7 134.8 263.4 559.2

625.8 (3.82)n (5.27) (4.01)

(16.89) (14.44) (18.41) (21.68)

(13.70) (16.32) (18.40) (21.68)

112.4 331.9 138.4 281.6

132.9 505.1 172.9 419.6

139.3 131.1 167.5 162.0

79.2 3321.2 (4.25)n (1.14)

190.6 559.6 2887.9 (4.23)

kroe100

3 4 5 6

11,323 10,019 9364 9035

343.4 919.5 1019.5 305.2

476.3 1103.0 1379.5 368.3

118.2 248.5 287.4 213.1

124.0 278.3 255.4 229.2

657.4 (4.79) (5.38)n (1.78)

(14.77) (21.25) (18.97) (21.69)

(15.36) (21.53) (17.75) (21.69)

228.3 1264.9 311.4 166.6

308.8 1750.2 451.3 238.0

179.2 274.2 166.8 99.9

308.9 3072.0 (4.28)n 2897.4

466.0 278.4 452.6 1260.5

rd100

3 4 5 6

4047 3652 3341 3234

594.2 3693.7 76.3 145.2

771.2 4432.9 88.4 158.7

116.4 157.8 84.4 143.3

116.1 167.6 76.0 140.8

2702.1 (4.24) (5.01) 6729.5

(17.11) (22.02) (19.60) (11.51)

(16.09) (21.83) (19.96) (11.51)

645.0 1561.1 140.3 41.0

983.6 2403.6 208.8 48.7

277.7 208.5 68.0 73.1

1773.5 (4.71)n 2480.5 632.4

528.0 6306.9 1849.5 2801.4

eil101

3 4 5 6

257 241 230 228

343.2 1098.1 624.4 554.0

427.0 1175.0 665.7 583.2

300.4 828.7 727.2 1669.8

298.4 899.0 693.9 1709.1

(1.71) (14.29)n (11.62) (11.32)

(41.21) (35.93) (26.93) (29.23)

(41.21) (35.93) (26.93) (29.23)

1090.4 1273.1 595.8 421.0

1368.5 2169.9 1043.6 634.1

924.5 1048.1 195.8 213.1

1110.5 (8.90)n (7.00)n (7.29)n

4034.8 (6.94)n (9.46) (13.61)

pr107

3 4 5 6

27,086 24,782 23,044 21,838

(4.96)n (8.13) (6.65) (3.61)

(5.20) (7.80) (6.73) (3.65)

374.0 2831.7 3269.6 4068.7

377.6 3029.7 3826.7 4653.6

55.4 3203.0 (1.61)n 1716.9

5028.2 (4.06) (9.29) (9.90)

2802.5 (3.94) (8.49) (9.77)

110.5 566.4 379.0 199.3

122.6 733.8 481.0 252.1

121.8 252.4 130.2 85.9

32.0 527.1 1511.1 1413.2

10.6 156.5 332.3 166.0

pr124

3 4 5 6

36,124 32,395 30,522 29,284

1503.6 6459.1 5748.9 1736.6

2200.3 6518.8 6965.1 1832.5

267.1 758.6 591.5 936.6

241.8 703.1 556.4 995.1

2998.8 (2.53) (7.16) (7.07)

(13.10) (19.46) (27.47) (13.48)

(14.03) (19.73) (27.46) (13.48)

312.8 1925.8 2647.2 303.2

411.3 3004.1 4029.1 500.5

196.4 321.7 455.2 88.1

1796.6 (8.86)n (5.48)n 7683.6

458.3 4464.3 2095.4 (4.18)n

bier127

3 4 5 6

65,227 60,944 59,934 59,544

(1.67) 2796.0 5078.0 7188.4

(2.96) 4585.7 5649.8 9406.0

(2.96) 5668.7 5038.9 4405.9

(2.73) 7636.5 7015.5 6119.1

8430.0 6075.9 (2.45)n (2.69)

(11.54) (15.28) (12.74) (10.68)

(7.34) (14.81) (12.74) (10.68)

565.6 162.2 146.3 110.3

825.9 215.0 188.5 147.1

430.8 247.2 271.9 149.3

1788.9 (2.70)n (2.14)n 4961.1

992.9 2083.0 9565.1 (1.32)

gr137

3 4 5 6

521 450 431 410

(12.69) (21.80) (17.52) (23.83)

(12.69) (21.80) (17.52) (23.83)

(1.48) (16.46) (12.82) (12.29)

(1.65) (16.41) (12.50) (11.81)

(24.58) (36.82) (35.86) (37.25)

(41.70) (40.83) (59.17) (56.58)

(41.71) (39.92) (59.43) (56.63)

(17.17) (19.30) (19.20) (15.63)

(15.27) (21.69) (19.03) (16.24)

(7.77) (7.84) (7.64) (6.82)

(17.72)n (22.97) (31.82)n (30.65)n

(11.72) (15.47) (19.49) (21.30)

pr144

3 4 5 6

44,716 40,665 37,666 36,903

(0.47) (4.57) (0.49) 2312.6

(1.27) (4.78) (0.48) 2100.9

760.4 6111.2 4318.1 2303.0

795.1 6055.9 4572.3 2108.3

1249.7 (7.77)n (5.26) (5.68)

(3.08) (31.51) (22.95) (17.65)

(3.39) (31.52) (22.96) (17.65)

132.9 2509.5 750.7 488.9

145.5 3690.0 772.3 582.2

167.1 536.2 469.4 419.1

110.5 (5.39)n (4.63)n (4.70)n

50.5 3710.5 (0.04) 4980.5

kroa150

3 4 5 6

13,064 11,588 10,805 10,565

(8.07) (6.13) (2.30) (0.42)

(7.85) (7.18) (2.52) (0.87)

6410.2 8248.1 4528.1 8011.7

7262.8 (0.49) 3924.1 (0.94)

(17.06)n (20.97) (20.15) (16.33)

(29.58) (35.12) (35.52) (31.67)

(29.37) (34.89) (35.51) (31.67)

(8.55) (6.17) 9423.6 3479.2

(9.22) (6.61) (1.64) 5011.3

(9.19) (2.43) 5153.1 5785.8

(15.89)n (15.74)n (12.27)n (14.31)n

(13.59)n (11.20)n (11.70) (13.65)

krob150

3 4 5 6

13,015 11,651 10,885 10,513

(3.42) (7.29) (5.97) (7.45)

(4.30) (7.29) (5.59) (4.19)

1507.4 4186.6 6913.5 8851.6

1557.2 4495.1 6418.3 8414.8

(16.35)n (24.22) (28.24) (28.65)

(34.21) (36.70) (43.67) (36.03)

(32.54) (36.70) (43.67) (36.03)

(6.06) (7.65) (4.90) (4.30)

(7.36) (10.21) (6.10) (4.62)

(1.46) (5.71) 7193.9 3677.5

(18.42)n (18.90)n (17.81)n (17.01)n

(13.41)n (14.18)n (15.76) (17.99)

pr152

3 4 5 6

53,602 44,870 42,279 40,402

(10.24) (7.95) (5.51) (2.68)

(11.08) (9.00) (5.57) (2.49)

4322.9 4807.1 6953.8 4046.1

3541.3 4518.1 7951.3 3825.1

(11.51)n (14.09)n (13.60) (14.85)

(19.80) (41.41) (31.76) (28.14)

(19.79) (40.84) (31.76) (28.14)

(6.68) (5.94) (4.03) (1.62)

(7.59) (6.21) (4.68) (2.08)

(0.58) 7876.3 9584.2 3570.2

(11.28)n (12.37)n (11.75)n (10.37)n

(3.13)n (2.09)n (6.23)n (5.44)

u159

3 4 5 6

24,306 22,055 20,062 19,157

(5.11) (12.14) (13.98) (7.27)

(5.12) (12.14) (13.98) (7.32)

3779.7 (4.89) (9.67) (2.35)

3381.7 (5.22) (9.19) (3.55)

(14.63) (34.33) (30.18) (39.59)

(34.64) (42.42) (39.80) (39.99)

(34.79) (42.41) (39.80) (39.99)

(7.86) (12.87) (9.45) (4.41)

(8.66) (14.89) (9.93) (5.39)

(1.47) (5.11) (0.75) 7494.5

(14.78)n (26.72)n (28.68)n (22.98)n

(13.94)n (19.27)n (19.52) (18.71)

16

M. Leitner / Computers & Operations Research 65 (2016) 1–18

Table 14 Detailed results for larger instances and grid clustering with μ ¼7. Besides best solution values (Best), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Best

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CL x;N

CLxþ

þ CLc; x

CLy

CL cy

kroa100

3 4 5 6

6716 6276 6154 6092

57.1 397.1 368.1 267.8

66.7 525.0 390.5 293.6

225.4 149.9 477.3 918.0

222.9 174.5 432.1 1037.7

613.1 4445.2 3981.5 5200.1

(38.97) (29.75) (31.67) (31.55)

(38.97) (29.75) (31.67) (31.55)

68.6 72.4 64.1 33.4

79.0 94.1 75.7 37.1

83.9 62.7 67.5 59.9

170.4 596.4 667.9 566.4

255.1 1290.7 1415.4 1074.2

krob100

3 4 5 6

6859 6506 6277 6217

16.3 105.2 95.4 189.3

16.5 117.3 100.2 183.2

15.6 116.2 97.9 308.1

16.0 104.8 104.8 294.8

111.8 2381.9 3064.6 8723.4

(22.20) (27.91) (25.93) (25.42)

(22.15) (27.91) (25.97) (25.42)

60.0 108.7 42.0 47.9

72.0 136.8 46.0 58.9

65.8 137.0 42.0 58.1

54.5 1020.1 933.9 491.1

46.6 224.2 980.0 4405.9

kroc100

3 4 5 6

6620 6056 5819 5722

65.0 124.3 135.2 131.0

86.0 136.9 140.4 133.2

41.0 86.6 151.2 128.7

41.5 87.7 148.4 132.3

317.4 1986.5 2453.1 (0.71)

(28.90) (22.57) (26.71) (23.14)

(28.90) (22.57) (26.71) (23.14)

95.9 55.0 32.5 29.2

124.1 71.0 36.5 33.4

96.5 41.8 37.8 30.2

96.5 212.1 221.5 120.6

120.6 120.5 124.5 552.1

krod100

3 4 5 6

6911 6388 6035 5934

44.1 239.1 130.8 187.4

51.3 247.8 123.4 178.5

44.8 129.7 139.0 165.1

42.2 149.7 125.9 151.2

332.3 (10.75)n 6909.9 6575.6

(17.86) (25.11) (28.61) (18.59)

(17.51) (25.18) (28.61) (18.59)

136.8 107.4 47.4 27.2

160.1 130.4 47.3 26.8

87.6 138.6 57.0 27.9

150.6 737.8 885.5 691.3

99.4 177.6 461.2 1549.8

kroe100

3 4 5 6

7974 7251 6919 6770

36.6 61.8 163.2 184.6

45.8 70.6 187.0 210.5

72.9 68.7 110.7 225.5

71.8 69.0 107.5 240.4

205.2 2417.5 5620.3 6103.5

(36.06) (33.33) (33.32) (26.80)

(36.06) (33.33) (33.32) (26.80)

60.9 94.5 45.2 33.4

73.3 114.2 58.0 40.0

58.4 65.7 45.6 35.8

94.5 160.8 440.6 877.8

60.0 498.8 525.9 2470.6

rd100

3 4 5 6

2805 2520 2411 2338

32.5 61.8 78.8 85.1

40.1 80.4 70.3 86.8

49.6 66.2 72.3 87.7

50.8 64.5 73.6 92.4

482.5 1394.6 5193.9 4379.7

(26.11) (32.03) (35.52) (30.60)

(22.51) (31.55) (35.52) (30.60)

101.6 70.1 48.6 16.3

135.3 89.8 60.5 16.2

85.5 75.0 44.2 14.6

269.5 429.4 967.6 192.0

221.5 255.7 337.4 563.9

eil101

3 4 5 6

161 148 145 144

431.4 139.2 851.1 365.6

491.1 142.7 860.4 459.3

345.4 685.5 930.0 1359.7

331.7 790.9 938.2 1702.3

2938.2 3476.3 6960.6 (8.40)

(35.47) (34.57) (26.35) (23.93)

(35.48) (34.57) (26.35) (23.93)

154.6 56.4 81.4 45.1

179.2 66.6 100.1 52.2

204.1 52.6 84.9 62.9

871.6 1609.2 1631.4 941.6

964.8 1177.8 3018.6 (7.64)

pr107

3 4 5 6

20,809 19,219 18,196 17,801

1801.6 1852.5 642.1 645.6

2024.0 1859.8 652.4 638.5

346.8 549.3 564.5 586.3

346.7 593.7 611.4 718.4

44.8 901.9 354.7 349.0

(6.45) (13.25) (11.67) (10.96)

(3.88) (13.44) (11.67) (10.96)

70.2 224.2 195.6 161.5

71.9 244.1 186.7 161.7

90.7 186.1 211.1 165.3

125.3 648.3 1601.5 649.9

51.2 363.8 1062.5 986.9

pr124

3 4 5 6

28,429 25,839 24,577 23,881

30.4 113.5 132.2 271.3

38.1 103.5 139.4 227.3

35.1 112.4 137.2 225.9

33.7 123.7 123.8 211.2

307.3 1131.6 1901.2 1441.5

(13.55) (14.76) (16.90) (9.16)

(13.36) (14.81) (16.90) (9.16)

49.8 82.8 64.0 34.9

59.6 97.1 74.9 39.7

42.6 48.8 49.8 22.9

311.2 609.5 2771.9 294.4

396.8 891.6 436.3 565.5

bier127

3 4 5 6

54,843 52,809 52,097 52,097

661.9 2248.6 2834.0 2882.4

1016.0 2482.7 3093.3 3322.7

765.3 2922.7 3384.5 4683.1

923.7 3645.5 3043.3 5272.3

398.6 882.0 7987.7 5462.4

(11.34) (14.89) (9.77) (8.94)

(6.72) (12.42) (9.77) (8.94)

154.3 103.0 80.3 64.0

160.3 108.1 86.2 66.8

133.3 228.9 79.5 86.5

68.3 283.5 (3.01)n 853.7

51.2 367.5 682.1 1907.0

gr137

3 4 5 6

363 323 300 287

(8.41) (11.16) (14.86) (10.14)

(8.33) (11.16) (14.86) (7.37)

7264.7 (3.07) (16.97) (3.18)

6955.1 (0.79) (17.46) (3.12)

(21.89)n (34.63) (32.73) (34.66)

(60.57) (54.83) (48.49) (38.73)

(60.63) (54.84) (48.49) (38.73)

(7.95) (9.94) (9.69) (5.17)

(9.59) (11.88) (8.32) (5.03)

(4.59) (5.88) (3.53) (2.95)

(17.09)n (34.69)n (33.89)n (29.76)n

(17.98)n (23.55)n (22.00)n (27.03)

pr144

3 4 5 6

40,511 37,750 35,212 34,626

(1.41) (4.44) 1980.8 2748.5

(1.62) (4.66) 2139.7 3149.6

362.3 2808.1 1736.1 1572.6

353.9 2245.3 1740.7 1750.3

517.6 (5.78)n (8.55) (5.28)

(9.92) (23.05) (16.91) (16.09)

(12.16) (22.78) (16.91) (16.09)

283.8 1425.5 296.3 216.0

340.4 1848.1 356.4 248.2

172.1 570.3 215.7 238.8

1515.4 (4.89)n (3.88)n (5.36)n

574.4 (5.54)n (1.64)n 7655.2

kroa150

3 4 5 6

9426 8801 8493 8248

1481.6 7161.7 6863.9 3662.6

1740.3 9186.3 7025.9 2830.6

905.6 1951.5 3641.5 4094.7

941.1 2197.4 3058.3 3773.9

(2.68) (17.96) (16.92) (19.19)

(33.12) (23.84) (29.02) (20.23)

(33.12) (23.84) (29.02) (20.23)

3422.1 3114.3 4070.9 1765.8

3507.6 3330.0 5125.9 2439.6

1410.9 831.7 1207.3 655.6

(5.67)n (14.57)n (15.41)n (11.79)n

8794.0 (7.91)n (12.87)n (13.56)

krob150

3 4 5 6

9365 8314 7965 7754

(3.60) (3.54) (3.79) (2.94)

(6.59) (5.61) (4.44) (3.65)

1558.5 3043.6 5536.6 6434.6

1601.3 3359.4 5834.3 4754.6

(22.13)n (22.56) (28.35) (33.55)

(37.84) (50.11) (39.23) (40.23)

(37.84) (50.11) (39.23) (40.23)

(4.02) (1.49) (2.14) 2931.0

(5.15) (1.92) (2.77) 4495.4

5909.0 1767.7 3714.1 4922.9

(22.31)n (22.14)n (16.48)n (22.31)n

(6.89)n (18.23)n (17.74) (24.27)

pr152

3 4 5 6

46,009 38,609 36,934 36,172

(8.88) (1.95) (0.95) 3798.3

(8.84) (1.94) (1.00) 3409.0

4526.4 4123.3 3216.8 3034.3

4848.7 3965.8 4137.5 3632.2

(3.94)n (9.34) (9.28) (12.03)

(26.02) (17.75) (18.86) (16.37)

(26.12) (17.75) (18.86) (16.37)

5130.1 1041.0 867.1 1113.0

6441.1 1177.5 950.8 1470.1

1449.3 660.3 637.2 836.8

2859.6 (8.72)n (8.82)n (7.41)n

(4.06)n (7.10)n (6.27)n (5.69)n

u159

3 4 5 6

16,948 15,589 14,540 13,793

1110.1 (8.65) (2.30) 6783.3

1021.2 (7.74) (2.31) 5534.0

758.1 7303.9 7458.0 6955.0

707.2 6747.8 8639.9 5948.6

(13.50)n (26.34) (29.49) (28.20)

(44.60) (43.22) (40.18) (41.09)

(44.60) (43.22) (40.18) (41.09)

6151.6 (4.45) (2.99) 2680.1

6358.6 (4.97) (3.90) 3384.8

2774.1 3935.2 1919.9 838.0

(9.93)n (24.83)n (19.15)n (24.13)n

(6.84)n (17.40)n (19.89)n (17.30)

M. Leitner / Computers & Operations Research 65 (2016) 1–18

17

Table 15 Detailed results for larger instances and grid clustering with μ¼10. Besides best solution values (Best), CPU-times in seconds are reported whenever an instance is solved to proven optimality by the corresponding method while values in parenthesis are optimality gaps in case of termination due to time or memory (indicated by n) limit. Inst.

H

Best

þ Lx;N

Lxþ

þþ Lc; x;L

þþ Lc; x

Ly

þ Lc; y;L

þ Lc; y

þ CLx;N

CL xþ

þ CLc; x

CLy

CLcy

kroa100

3 4 5 6

6716 6276 6154 6092

54.9 398.1 361.9 274.8

63.9 535.9 411.4 288.1

213.6 145.5 416.8 1065.7

228.3 184.4 455.8 1004.6

618.6 4661.6 4348.3 5861.6

(38.97) (29.75) (31.67) (31.55)

(38.97) (29.75) (31.67) (31.55)

66.9 72.7 63.4 33.0

78.0 94.3 75.8 36.5

83.2 62.1 66.0 60.6

166.5 598.1 640.8 572.3

251.4 1282.7 1413.7 1043.0

krob100

3 4 5 6

6859 6506 6277 6217

15.0 101.9 93.9 195.1

17.0 120.5 100.0 191.5

15.0 102.7 98.8 300.5

15.2 97.1 96.4 282.8

114.5 2547.7 3191.3 9091.5

(22.14) (27.91) (25.93) (25.42)

(22.14) (27.91) (25.93) (25.42)

62.5 109.4 41.5 46.1

71.6 136.9 47.3 59.0

61.9 135.6 41.4 56.5

55.4 971.9 879.4 478.3

46.1 230.0 967.1 4449.2

kroc100

3 4 5 6

6620 6056 5819 5722

65.7 125.0 130.1 127.5

86.5 134.0 133.4 135.9

40.9 87.6 150.2 126.8

41.3 86.5 145.6 126.3

312.3 1997.2 2506.8 (1.44)

(28.90) (22.57) (26.71) (23.14)

(28.90) (22.57) (26.71) (23.14)

95.5 56.0 32.6 29.3

125.6 70.4 37.6 31.9

92.5 42.4 37.9 28.7

96.2 211.7 216.3 118.6

116.8 119.2 124.5 559.9

krod100

3 4 5 6

6911 6388 6035 5934

45.0 238.4 125.2 184.6

50.0 251.6 132.6 195.8

42.7 131.6 138.0 159.9

42.2 131.9 144.5 169.1

339.7 (10.75)n 7331.4 7085.8

(17.86) (25.42) (28.61) (18.59)

(17.51) (25.15) (28.61) (18.59)

141.0 107.2 45.7 26.2

160.0 129.5 48.4 26.3

86.7 133.0 57.4 26.8

148.0 740.3 906.0 686.3

99.5 172.5 474.1 1540.3

kroe100

3 4 5 6

7974 7251 6919 6770

36.1 63.9 161.6 193.1

45.1 73.5 198.5 223.9

72.1 74.8 107.2 240.6

73.6 70.8 116.5 237.9

215.1 2645.0 5841.7 6044.8

(36.06) (33.33) (33.32) (26.80)

(36.06) (33.33) (33.32) (26.80)

61.1 94.6 45.0 35.5

73.3 111.7 58.5 40.1

59.3 64.8 46.0 36.0

90.6 162.7 462.0 844.6

59.2 501.1 548.3 2525.3

rd100

3 4 5 6

2805 2520 2411 2338

31.2 66.0 68.5 89.8

39.3 71.0 73.1 72.0

48.9 71.0 82.8 82.5

49.0 67.7 80.3 82.7

477.3 1359.7 5243.3 3366.0

(26.11) (31.53) (35.52) (30.60)

(22.51) (32.02) (35.52) (30.60)

100.6 69.8 49.3 15.8

136.4 90.4 61.2 15.9

84.4 74.6 44.2 14.8

271.1 418.3 964.2 193.1

223.0 253.2 312.3 557.3

eil101

3 4 5 6

161 148 145 144

399.8 132.6 824.3 421.2

474.8 145.5 880.3 411.9

338.6 653.8 1036.9 1712.8

347.7 759.6 771.6 1750.0

2978.8 3680.6 7402.5 (8.61)

(35.47) (34.57) (26.35) (23.93)

(35.48) (34.57) (26.35) (23.93)

155.8 56.7 82.7 46.2

175.3 65.4 99.8 50.4

207.1 51.6 79.0 64.2

786.0 1639.0 1685.7 955.9

985.4 1175.0 2832.0 (7.82)

pr107

3 4 5 6

18,939 17,542 16,802 16,756

202.9 171.8 300.6 579.9

227.6 170.4 314.7 587.3

191.3 172.1 301.4 541.3

193.9 170.3 311.3 542.4

58.2 62.9 43.2 127.1

(9.02) (10.65) (7.48) (7.57)

(5.55) (10.65) (7.48) (7.57)

19.2 75.5 223.8 158.2

19.4 75.3 236.6 151.4

18.6 74.8 232.5 152.2

40.8 117.8 136.8 167.4

71.7 140.8 89.0 140.8

pr124

3 4 5 6

19,953 18,682 18,600 18,554

39.5 43.5 163.6 211.1

59.5 50.5 208.0 201.4

59.8 100.8 225.1 302.8

63.1 115.7 269.7 325.9

94.9 185.6 287.1 606.1

(18.04) (22.38) (18.42) (18.85)

(10.31) (22.37) (18.42) (18.85)

21.9 11.1 15.5 16.0

27.1 12.2 16.5 18.1

26.4 12.2 19.3 17.7

45.2 82.5 200.7 315.6

48.8 88.0 258.5 953.3

bier127

3 4 5 6

44,909 43,825 43,778 43,778

761.8 574.7 833.3 582.1

1573.9 934.6 1309.9 867.0

1007.3 1237.4 1414.1 1859.1

1243.3 1035.8 1864.0 2307.0

78.9 360.3 1723.0 543.1

(10.46) (13.30) (12.63) (12.55)

3439.9 (12.61) (12.80) (12.33)

49.9 21.0 23.0 30.7

60.8 23.7 29.8 41.0

59.6 25.6 26.3 37.2

57.5 48.5 85.7 186.8

41.5 79.8 94.1 181.8

gr137

3 4 5 6

261 232 215 208

(7.53) (6.70) (3.76) (4.85)

(7.37) (7.40) (3.40) (5.60)

3107.4 3876.1 2532.0 3602.9

3072.5 4017.3 2764.7 3996.6

(13.63)n (18.24) (23.09) (26.34)

(46.20) (48.42) (49.41) (48.48)

(46.05) (48.42) (49.41) (48.48)

3422.0 3413.3 1619.6 1675.5

3723.6 3293.5 2153.7 2108.0

4175.1 1662.7 652.2 592.2

(19.28)n (22.69)n (30.29)n (26.58)n

(15.85)n (23.88)n (25.45)n (27.74)n

pr144

3 4 5 6

37,199 34,497 33,741 33,021

1622.4 548.6 2279.7 323.5

1963.8 572.1 2210.5 280.8

204.5 433.1 845.9 379.9

205.1 397.7 775.8 439.7

112.4 865.3 2538.4 809.0

(5.44) (9.12) (12.21) (6.67)

(5.40) (9.12) (12.21) (6.67)

57.1 122.7 122.0 45.4

60.1 125.2 130.9 45.5

44.0 134.4 120.1 54.0

88.0 237.8 2322.6 697.4

77.9 260.2 910.9 1200.0

kroa150

3 4 5 6

6016 5574 5413 5337

546.9 949.8 2374.3 9787.2

636.8 980.3 2180.8 8888.5

1614.2 7961.1 2535.7 3961.3

1344.6 9321.4 3288.5 4016.9

4360.2 (12.59) (22.70) (23.11)

(41.78) (38.96) (37.29) (42.28)

(41.78) (38.96) (37.29) (42.28)

284.2 534.0 310.6 130.3

336.3 719.4 341.6 156.5

237.3 369.1 273.8 113.0

1349.3 (11.24)n (10.22)n 2969.2

1457.3 (15.32)n 5289.7 (12.82)n

krob150

3 4 5 6

6390 5889 5743 5668

705.8 1661.6 1447.5 1184.1

959.2 2053.2 1420.9 1276.1

558.7 1072.7 981.7 4099.6

457.2 866.0 1134.3 2956.4

(11.02)n (16.47) (17.16) (18.95)

(37.88) (45.88) (37.67) (36.08)

(37.88) (45.88) (37.67) (36.08)

470.8 410.5 425.9 218.3

517.9 450.8 470.4 235.4

439.8 302.1 314.3 245.0

1392.4 (8.25) (11.11)n (12.48)n

2119.8 (9.97)n (17.35)n (18.92)n

pr152

3 4 5 6

38,206 34,874 33,785 33,479

4115.0 2028.8 3249.8 4045.8

4198.1 2890.1 3156.5 4548.2

833.3 930.1 2446.0 5727.1

780.2 1019.2 2547.7 5876.4

3042.8 4166.4 (0.60) (12.52)

(16.93) (13.70) (18.93) (16.31)

(16.93) (13.70) (18.93) (16.31)

868.2 170.0 442.5 186.8

922.3 175.4 471.8 193.4

477.2 184.8 408.8 214.7

(5.16)n (3.54)n (7.77)n (6.98)n

3946.9 (4.94)n (4.33)n (6.95)n

u159

3 4 5 6

16,948 15,589 14,540 13,793

994.1 (6.72) (1.58) 5041.7

1437.4 (6.86) (1.86) 5425.1

620.3 6249.8 7105.5 5721.7

715.6 6498.3 5996.4 5855.9

(13.50)n (25.11) (28.47) (27.12)

(44.60) (43.22) (40.18) (41.09)

(44.60) (43.22) (40.18) (41.09)

5117.0 (2.84) (1.48) 3431.2

5705.7 (3.53) (3.35) 3081.4

2469.5 3828.4 1998.1 643.9

(9.93)n (24.83)n (19.15)n (24.13)n

(6.84)n (17.40)n (19.89)n (16.06)

18

M. Leitner / Computers & Operations Research 65 (2016) 1–18

References [1] Botton Q, Fortz B, Goueia L, Poss M. Benders decomposition for the hopconstrained survivable network design problem. INFORMS J Comput 2013;25 (1):13–26. [2] Cherkassky BV, Goldberg A. On implementing push-relabel method for the maximum flow problem. Algorithmica 1994;19:390–410. [3] Dahl G. The 2-hop spanning tree problem. Oper Res Lett 1998;23(1–2):21–6. [4] Dahl G, Gouveia L, Requejo C. On formulations and methods for the hopconstrained minimum spanning tree problem. In: Resende MGC, Pardalos PM, editors. Handbook of optimization in telecommunications. US: Springer; 2006. p. 493–515. [5] Corinne Feremans. Generalized spanning trees and extensions [Ph.D. thesis]. Universite Libre de Bruxelles, 2001. [6] Feremans C, Labbé M, Laporte G. The generalized minimum spanning tree problem: polyhedral analysis and branch-and-cut algorithm. Networks 2004;43(2):71–86. [7] Fischetti M, José J, González S, Toth P. A branch-and-cut algorithm for the symmetric generalized traveling salesman problem. Oper Res 1997;45 (3):378–94. [8] Golden B, Raghavan S, Stanojević D. Heuristic search for the generalized minimum spanning tree problem. INFORMS J Comput 2005;17(3):290–304. [9] Gouveia L. Using the Miller–Tucker–Zemlin constraints to formulate a minimal spanning tree problem with hop constraints. Comput Oper Res 1995;22:959–70. [10] Gouveia L. Multicommodity flow models for spanning trees with hop constraints. Eur J Oper Res 1996;95:178–90. [11] Gouveia L, Requejo C. A new Lagrangian relaxation approach for the hopconstrained minimum spanning tree problem. Eur J Oper Res 2001;132 (3):539–52. [12] Gouveia L, Magnanti TL, Requejo C. An intersecting tree model for odddiameter-constrained minimum spanning and Steiner trees. Ann Oper Res 2006;146(1):19–39. [13] Gouveia L, Paias A, Sharma D. Restricted dynamic programming based neighborhoods for the hop-constrained minimum spanning tree problem. J Heuristics 2011;17(1):23–37. [14] Gouveia L, Simonetti L, Uchoa E. Modeling hop-constrained and diameterconstrained minimum spanning tree problems as Steiner tree problems over layered graphs. Math Progr 2011;128:123–48. [15] Gouveia L, Leitner M, Ljubić I. Hop constrained Steiner trees with multiple root nodes. Eur J Oper Res 2014;236:100–12. [16] Gouveia L, Leitner M, Ljubić I. The two-level diameter constrained spanning tree problem. Math Progr 2015;150(1):49–78. http://dx.doi.org/10.1007/ s10107-013-0743-z.

[17] Bin Hu. Hybrid Metaheuristics for generalized network design problems [Ph.D. thesis]. Vienna University of Technology, 2008. [18] Hu B, Leitner M, Raidl GR. Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem. J Heuristics 2008;14(5):473–99. [19] Hu B, Leitner M, Raidl GR. The generalized minimum edge biconnected network problem: efficient neighborhood structures for variable neighborhood search. Networks 2010;55(3):257–75. [20] Karapetyan D, Gutin GZ. Lin–Kernighan heuristic adaptations for the generalized traveling salesman problem. Eur J Oper Res 2011;208(3):221–32. [21] Koch T, Martin A. Solving Steiner tree problems in graphs to optimality. Networks 1998;32(3):207–32. [22] LeBlanc LJ, Chifflet J, Mahey P. Packet routing in telecommunication networks with path and flow restrictions. INFORMS J Comput 1999;11(2):188–97. [23] Ljubić I, Gollowitzer S. Layered graph approaches to the hop constrained connected facility location problem. INFORMS J Comput 2013;25:256–70. [24] Myung Y-S, Lee C-H, Tcha D-W. On the generalized minimum spanning tree problem. Networks 1995;26(4):231–41. [25] Öncan T, Cordeau J-F, Laporte G. A tabu search heuristic for the generalized minimum spanning tree problem. Eur J Oper Res 2008;191(2):306–19. [26] Petrica C. Pop. The generalized minimum spanning tree problem [Ph.D. thesis]. University of Twente; 2002. [27] Pop PC, Kern W, Still. G. A new relaxation method for the generalized minimum spanning tree problem. Eur J Oper Res 2006;170(3):900–8. [28] Reinelt G. TSPLIB—a traveling salesman problem library. INFORMS J Comput 1991;3(4):376–84. [29] Mario Ruthmair. On solving constrained tree problems and an adaptive layers framework [Ph.D. thesis]. Vienna University of Technology; 2008. [30] Ruthmair M, Raidl GR A layered graph model and an adaptive layers framework to solve delay-constrained minimum tree problems. In: Proceedings of the 15th Conference on Integer Programming and Combinatorial Optimization (IPCO XV), vol. 6655 of LNCS; 2011. p. 376–388. [31] John Silberholz, Bruce Golden. The generalized traveling salesman problem: a new genetic algorithm approach. In: Edward K Baker, Anito Joseph, Anuj Mehrotra, Michael A. Trick, editors. Extending the horizons: advances in computing, optimization, and decision technologies. Operations research/ computer science interfaces series, vol. 37; 2007. p. 165–81. [32] Voß S. The Steiner tree problem with hop constraints. Ann Oper Res 1999;86:271–94. [33] Woolston KA, Albin S. The design of centralized networks with reliability and availability constraints. Comput Oper Res 1988;15:207–17.