Min-degree constrained minimum spanning tree problem with fixed centrals and terminals: Complexity, properties and formulations

Accepted Manuscript

Min-Degree Constrained Minimum Spanning Tree Problem with Fixed Centrals and Terminals: complexity, properties and formulations Fabio C.S. Dias, Manoel Campelo, ˆ Cr´ıston Souza, Rafael Andrade PII: DOI: Reference:

S0305-0548(17)30060-6 10.1016/j.cor.2017.03.001 CAOR 4206

To appear in:

Computers and Operations Research

Received date: Revised date: Accepted date:

5 September 2016 18 January 2017 7 March 2017

Please cite this article as: Fabio C.S. Dias, Manoel Campelo, Cr´ıston Souza, Rafael Andrade, ˆ Min-Degree Constrained Minimum Spanning Tree Problem with Fixed Centrals and Terminals: complexity, properties and formulations, Computers and Operations Research (2017), doi: 10.1016/j.cor.2017.03.001

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Highlights • A variant of the Min-Degree Constrained Minimum Spanning Tree Problem is proposed • Each node is fixed as central (with a minimum degree constraint) or terminal (leaf) • The feasibility problem is shown to be NP-Complete • Necessary and sufficient conditions for feasibility are derived

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• Several ILP formulations and three Lagrangian heuristics are presented and tested

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Min-Degree Constrained Minimum Spanning Tree Problem with Fixed Centrals and Terminals: complexity, properties and formulations Fabio C. S. Diasb , Manoel Campˆeloa , Cr´ıston Souzab , Rafael Andradea de Estat´ıstica e Matem´ atica Aplicada, Universidade Federal do Cear´ a, Brasil b Universidade Federal do Cear´ a, Campus de Quixad´ a, Brazil

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a Departamento

Abstract

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We consider a variant of the Min-Degree Constrained Minimum Spanning Tree Problem where the central and terminal nodes are fixed a priori. We prove that the optimization problem is NP-Hard even for complete graphs and the feasibility problem is NP-Complete even if there is an edge between each central and each terminal in the input graph. Actually, this complexity result still holds when the minimum degree of each central node is restricted to be a same value d ≥ 2. We derive necessary and sufficient conditions for feasibility. We present several integer linear programming formulations - based on known formulations for the minimum spanning tree problem - along with a theoretical comparison among the lower bounds provided by their linear relaxations. We propose three Lagrangian heuristics. Computational experiments compare the performances of the heuristics and the formulations.

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Keywords: Min-Degree Constrained Minimum Spanning Tree Problem, Integer Programming, Lagrangian Heuristic.

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1. Introduction

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A tree is a minimally connected graph or, more precisely, an acyclic connected graph. A spanning tree of a graph is a tree containing all its vertices and some of its edges. If the graph models a network, then a spanning tree comprises a minimal subset of links which allows communication between any pair of nodes of the network; or still, it is a substructure of the network that guarantees connectivity with no redundant links. Actually, there exists a wide range of applications that can be modeled by a spanning tree or a spanning tree satisfying additional constraints [17]. Given a weighted graph G = (V, E) with vertex set V , edge set E and cost P ce ∈ R associated with each edge e ∈ E, the cost of a spanning tree H = (V, E 0 ) is c(H) = e∈E 0 ce . The basic problem in this context, to be denoted MST, is that of finding a minimum-cost spanning tree. The literature considers several variants of this fundamental problem, where the desired spanning tree H has to verify some extra conditions besides being connected and acyclic [1, 14, 15, 27]. One of these conditions is related to the degree dH (v) of each vertex v ∈ V , i.e. the number of edges incident to v in H [2, 23]. Email addresses: [email protected] (Fabio C. S. Dias), [email protected] (Manoel Campˆ elo), [email protected] (Cr´ıston Souza), [email protected] (Rafael Andrade )

Preprint submitted to Elsevier

March 8, 2017

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Regarding this parameter, we introduce here a new variant of MST, to be called Min-Degree Constrained Minimum Spanning Tree with Fixed Centrals and Terminals (FMD-MST). In this case, we partition the set of vertices into V = C ∪ T , C ∩ T = ∅. The vertices in C are called centrals and have to satisfy a minimum degree constraint; the vertices in T are called terminals and must have degree 1.

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Problem 1 (FMD-MST). Let G = (C ∪ T, E) be a connected simple graph with cost ce ∈ R for each edge e ∈ E, and let dv ∈ Z, dv ≥ 1, be a minimum degree value for each vertex v ∈ C. Find a minimum-cost spanning tree H of G such that dH (v) ≥ dv for each v ∈ C, and dH (v) = 1 for each v ∈ T .

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A minimum-degree constraint in a spanning tree can represent, for example, a minimum number of connections that a hub or distribution center should have in order to be cost-effective (when the grounds for justifying its installation or operation is a threshold on the number of served nodes). On the other hand, it is natural the existence of end-users that must be served as peripheral nodes. Thus, problem FMD-MST can model applications from network design in telecommunications and distribution systems. Besides, it encompasses theoretical questions regarding the existence of spanning trees satisfying some constraints, e.g. when a subset (dv = 1, ∀v ∈ C) or the whole set (dv = 2, ∀v ∈ C) of the leaves is specified or, as we will see, when the non-leaf nodes have a maximum degree constraint. In this context, we consider the feasibility problem associated with FMD-MST. It can be seen as the FMD-MST with unit costs. Indeed, in the case of unit costs any feasible solution has the same value, since the number of edges in a tree is always one less than the number of vertices.

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Problem 2 (FMD-ST). Let G = (C ∪ T, E) be a connected simple graph, and let dv ∈ Z, dv ≥ 1, be a minimum degree value for each vertex v ∈ C. Is there a spanning tree H of G such that dH (v) ≥ dv for each v ∈ C, and dH (v) = 1 for each v ∈ T ?

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It is worth remarking some relations between these problems and the minimum spanning tree problem. First, observe that FMD-MST generalizes MST, which comprises the case where T = ∅ and dv = 1 for all v ∈ C. On the other hand, a feasible solution of FMD-MST is formed by a spanning tree of the subgraph G(C) of G induced by the centrals together with a link from each terminal to a central. Note, however, that a minimum spanning tree of G(C) may not be part of an optimal (or even feasible) solution of FMD-MST. See Figure 1 for an example. v2

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Figure 1: An instance of FMD-MST: C = {v1 , v2 , v3 , v4 }, T = {v5 , v6 }, dv = 2 for all v ∈ C; cost of each edge is 0 except for c(v3 v4 ) = 1 and c(v3 v6 ) = 2. (a) Graph G, and the MST of G(C) in thicker lines. (b) Optimal solution of FMD-MST. (c) Unique feasible solution using the MST of G(C). Removing edge v4 v5 from G yields no feasible solution using the MST of G(C).

1.1. Related work Besides its own relevance, the study of FMD-MST is motivated by its close relation with two other problems from the literature, namely the Min-Degree Constrained Minimum Spanning Tree 3

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Problem (MD-MST) and the Degree-Constrained Minimum Spanning Tree Problem (DC-MST). Both problems, defined below, consider constraints on the vertex degrees. Problem 3 (MD-MST). Let G = (V, E) be a connected simple graph with cost ce ∈ R for each edge e ∈ E, and let dv ∈ Z, dv ≥ 1, be a minimum degree value for each vertex v ∈ V . Find a minimum-cost spanning tree H of G such that, for all v ∈ V , dH (v) ≥ dv or dH (v) = 1.

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The main difference between FMD-MST and MD-MST is that, in the last problem, the central and terminal vertices are not defined a priori. Any vertex can be central or terminal. So, FMD-MST is a restriction of it. Problem MD-MST was introduced in [2], where it was shown to be NP-hard even if dv = d ≥ 3 for all v ∈ V . Note that it is equivalent to MST when dv ≤ 2 for all v ∈ V . Starting from that pioneer paper, several computational studies have been presented (see e.g. [3, 19, 20]). Being a restriction of MD-MST, FMD-MST could provide upper bounds for that problem. Unfortunately, even FMD-ST is NP-Complete, as we show in the next section. Problem 4 (DC-MST). Let G = (V, E) be a connected simple graph with cost ce ∈ R for each edge e ∈ E, and let dv ∈ Z+ , dv ≤ n − 1, be a maximum degree value for each vertex v ∈ V . Find a minimum-cost spanning tree H of G such that dH (v) ≤ dv for all v ∈ V .

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Problem DC-MST is an extensively studied NP-hard combinatorial optimization problem [13]. There is a vast literature about it, specially regarding solution methods (see e.g. [4, 5, 7, 23, 26, 28]). We show that DC-MST can be reduced to FMD-MST. This work contributes to the study both theoretical and algorithmic of the proposed problem without disregarding the related literature. First, we investigate the computational complexity of FMD-MST. The basic NP-Completeness results are in connection with those of MD-MST and DC-MST. As an extension, the study of the feasibility problem brings about new insights. The derived characterization of feasibility leads to the identification of more restricted NP-complete cases as well as polynomial cases, which are expressed in terms of the graph structure or the degree requirements. From the theoretical findings, we consider some solution methods and carry out extensive computational experiments. We test the counterparts for FMD-MST of the most successful integer linear programming (ILP) formulations proposed for MD-MST and DC-MST [2, 3, 4, 7, 20, 21]. Besides, we propose effective Lagrangian algorithms. We can anticipate that good computational performances were obtained by some of the proposed solution methods in hard instances of MD-MST (with an a priori partition of centrals and terminals). This may indicate a strategy for using FMD-MST as a subproblem to solve MD-MST, for example within a Benders decomposition framework.

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1.2. Main Notation Throughout the text we adopt the following notation and assumptions. For a simple graph G, V (G) and E(G) denote its vertex set and its edge set, respectively. Let V 0 , V 00 ⊆ V (G). The subset of edges of G with one endpoint in V 0 and the other endpoint in V 00 is δG (V 0 : V 00 ) = {uv ∈ E(G) : u ∈ V 0 , v ∈ V 00 }. We may simply use δG (V 0 ) = δG (V 0 : V \ V 0 ) and δG (v) = δG ({v}). The degree of a vertex v in G is dG (v) = |δG (v)|, and ∆(G) = max{dG (v) : v ∈ V }. The subgraph of G induced by V 0 is G(V 0 ) = (V 0 , E(V 0 )), where E(V 0 ) = δG (V 0 : V 0 ). Similar definitions apply for a digraph D, where there is a set A of arcs (oriented edges) instead of the set E of edges. − + (V 0 ) = δD (V \ V 0 : V 0 ). If Due to the orientation, we now use δD (V 0 ) = δD (V 0 : V \ V 0 ) and δD the graph is clear by the context, we remove the subscript in the notation. When referring to FMD-MST or FMD-ST, we consider G = (C ∪ T, E) as a simple, connected graph, where C and T are the sets of centrals and terminals, respectively. The number of centrals, terminals, vertices, and edges in G are denoted by c = |C|, t = |T |, n = c + t, and m = |E|, 4

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respectively. For each vertex v ∈ C, dv denotes the required minimum degree of v. When all centrals have the same minimum degree value (to be generally denoted d), we call the problem d-uniform. We assume that dG (v) ≥ dv for all v ∈ C, since otherwise the problem is infeasible. 1.3. Outline

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The paper is organized as follows. Section 2 is devoted to study the computational complexity of FMD-MST and FMD-ST. We prove that the feasibility problem is NP-complete and the optimization problem is NP-Hard even in the d-uniform case, for every d ≥ 2. Actually, the NP-Hardness result holds even if the input graph is complete. The 1-uniform case is shown to be polynomial. In Section 3, we further study the feasibility problem. We derive necessary and/or sufficient conditions for feasibility, prove that the d-uniform case remains NP-Complete even if δG (C : T ) induces a complete bipartite graph, and show some polynomial cases (including the one where G is complete). Based on known ILP formulations for MST, we propose several formulations for FMD-MST in Section 4. A theoretical comparison among the lower bounds provided by their linear relaxations is carried out. Starting from a Lagrangian relaxation, we propose three algorithms for FMD-MST in Section 5. We use the Deflected Subgradient Method to solve the Lagrangian dual and embed some procedures to find and improve feasible solutions, thus ending with lower and upper bounds. In Section 6, we summarize computational results from extensive experiments carried out with the formulations and the Lagrangian algorithms. We use test instances from the literature (adapted from instances of MD-MST) as well as additional randomly generated instances. We compare the performances specially in terms of solution gaps and processing times. 2. Complexity

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In this section we prove that FMD-MST is NP-Hard. For this we show that FMD-ST is NPComplete via a reduction from the Hamiltonian Path Problem. We first consider the case in which the minimum degree value is uniform for all central nodes, that is, dv = d for all v ∈ C. In addition, we start assuming that the value of d is predefined and is not part of the input. We denote the corresponding problems by FMD-STd and FMD-MSTd . Recall that, if H is a tree, then |E(H)| = |V (H)| − 1 and so X dH (v) = 2|E(H)| = 2|V (H)| − 2. (1)

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In particular, a feasible solution H of FMD-ST satisfies X X dH (v) = t and dH (v) = (2n − 2) − t = 2c + t − 2. v∈T

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Theorem 1. For all d ≥ 2, FMD-STd is NP-Complete and FMD-MSTd is NP-Hard.

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Proof. FMD-STd belongs to NP, since we can check in polynomial time whether the certificate (a subset of edges) forms a spanning tree, and whether all central nodes have degree at least d. ˆ = (Vˆ , E) ˆ and r, s ∈ Vˆ be an instance of the problem of deciding whether there exists Let G ˆ between r and s). To turn this instance into an instance of FMD-STd , a Hamiltonian path (in G ˆ as the subgraph induced by the central nodes. In addition, for each vertex we consider G v ∈ Vˆ \ {r, s}, we create d − 2 terminals connected by edges only to v. We also create d − 1 terminals connected only to r, and d − 1 terminals connected only to s. The resulting graph G 5

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ˆ + (d − 2)|Vˆ | + 2 edges, and therefore has polynomial size has |Vˆ | + (d − 2)|Vˆ | + 2 nodes and |E| ˆ See Figure 2. with respect of the size of G. ˆ between r and s together with the created edges induce Clearly, any Hamiltonian path in G a solution of FMD-STd . On the other hand, by (2), the sum of degrees of the centrals in every feasible solution of this instance of FMD-STd is 2|Vˆ | + ((d − 2)|Vˆ | + 2) − 2 = d|Vˆ |. As each central node has minimum degree d, we conclude that each central node has degree exactly d in all feasible solutions for this instance. Thus, considering only the edges of the solution induced by the central nodes, we conclude that the nodes r and s have degree 1, and the remaining central ˆ nodes have degree 2. Therefore, these edges form a Hamiltonian path between r and s in G, since the solution is connected. Since deciding whether there is a Hamiltonian path in a general graph is an NP-Complete problem, the desired result follows. 

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Two observations can be made in connection with the above theorem. First, note that FMD-MSTd is NP-Hard even if G is complete. Indeed, if G is not complete, we can add the missing edges with a very high cost M (e.g. M greater than the n − 1 most costly edges) so that any feasible solution of the modified instance that includes a “dummy” edge is worse than a feasible solution of the original instance, if any. Therefore, if the problem in the complete graph is infeasible or has cost at least M , the original problem is infeasible. Otherwise, they have the same optimal solution. In addition, observe that any instance of FMD-STd can be polynomially transformed into an equivalent instance of FMD-ST. If d > n, we can return a trivial infeasible instance. Otherwise, it is just a matter of including in the input the c minimum degree values, each one stored in O(log n) bits. These two remarks lead to the following corollary. Corollary 1. FMD-ST is NP-Complete. FMD-MST is NP-Hard, even if G is complete.

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We now consider the case where d = 1. First, note that FMD-MST1 is a relaxation of FMD-MST. Indeed, in the first problem, the degree constraints of the centrals are trivially implied by the connectivity of the solution. Thus, FMD-MST1 is FMD-MST without those constraints. Moreover, this relaxation is easy to solve. Proposition 1. For d = 1, FMD-MSTd is polynomial.

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Proof. An optimal solution of FMD-MST1 is given by a minimum spanning tree of G(C) together with an edge between each terminal i ∈ T and a central i? ∈ arg minj {cij : ij ∈ δ(T : C)}. 3. Feasibility Recall that an instance of FMD-ST is defined by a graph G = (C ∪ T, E) and a set of minimum degree values dv , for v ∈ C. In the previous section, we studied the complexity of the problem as a function of the parameter d = dv , for v ∈ C, and proved that even the feasibility problem is NP-Complete, for every value d ≥ 2. In this section, we study more deeply the feasibility 6

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problem. We turn our attention to the influence of other input parameter (the graph G). We analyze how the partition into centrals and terminals or the structure of G(C) affect feasibility. The next proposition shows that feasibility requires a minimum number tmin of terminals, given by X tmin = (dv − 2) + 2, v∈C

whose expression can be simplified in the d-uniform to

Proposition 2. If FMD-MST is feasible, then t ≥ tmin .

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tmin = c(d − 2) + 2 = cd − 2(c − 1). P Proof. In anyP feasible solution H for FMD-MST we t − 1) and P Phave that v∈V dH (v) = 2(c +P d (v) ≥ d + t. Thus, 2(c + t − 1) ≥ d + t, which results in t ≥ v∈V H v∈C v v∈C v v∈C dv − 2c + 2. 

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In order to understand why the condition t ≥ tmin is not sufficient, let us analyze the duniform case. Let H be a feasible solution of FMD-MST. Since H(C) is necessarily a spanning tree of G(C), note that 2(c − 1) is the contribution of the edges in H(C) to the degree of the nodes in C. As tmin = cd − 2(c − 1), we conclude that tmin is the minimum amount of terminals required to produce a feasible solution when each vertex in C has degree at most d in H(C). Since the degree of nodes in H(C) could be greater than d, the number of terminals must be sufficient to P compensate the sum of these excesses. Precisely, we need t ≥ tmin + v∈C max{0, dH(C) (v) − d} terminals. To illustrate this situation, consider the example where d = 2 and G(C) is the tree of Figure 3. Of course, this tree is part of any feasible solution. Clearly, tmin = 2 terminals are not enough to complete the minimum degree of the leaves of G(C). Note that there are two nodes with degree > 2, each of them with excess of 1 unit. It follows that t ≥ tmin + 2 = 4 is needed.

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Figure 3: Subgraph G(C). Example where more than tmin terminals are needed.

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Besides having enough terminals, we need to be able to connect then to the central nodes with degree less than d in H(C). In the example of Figure 3, we need a matching between the 4 terminals and the central nodes with degree 1. In general, a necessary and sufficient condition for the feasibility of FMD-MST can be given in terms of the spanning trees of G(C) as follows. Theorem 2. The following assertions are equivalent:

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(i) FMD-MST is feasible; P (ii) there is a spanning tree H of G(C) such that v∈S (dv − dH (v)) ≤ |T (S)|, for all S ⊆ C; P (iii) there is a spanning tree H of G(C) such that v∈S (dH (v) − dv ) ≤ |T (C \ S)| − tmin , for all S ⊆ C.

where T (S) = {u ∈ T : δ(u : S) 6= ∅} is the subset of terminals with at least one edge incident to S ⊆ C in G. 7

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P Proof. Let H be a spanning tree of G(C). Note that v∈S (dv − dH (v)) ≤ |T (S)|, ∀S ⊆ C if, and only if, X (dv − dH (v)) ≤ |T (S)|, ∀S ⊆ CH := {v ∈ C : dH (v) < dv }. (3) v∈S

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Indeed, in (ii), the inequality defined by S is weaker than the one defined by S ∩ CH . Therefore, we can replace the condition in (ii) by (3). Associated with H, define the bipartite graph BH obtained from the subgraph induced by the edges δ(CH : T ) where each vertex v ∈ CH is split into dv − dH (v) > 0 copies. Precisely, V (BH ) = {v i : v ∈ CH , i = 1, 2, . . . , dv − dH (v)} ∪ T (CH ) and E(BH ) = {v i u : vu ∈ δ(CH : T ), i = 1, 2, . . . , dv − dv (H)}. (i) ⇔ (ii) FMD-MST is feasible if, and only if, there is a spanning tree of G(C) such that the minimum degree of every vertex v in CH can be completed by connecting dv − dH (v) terminals from T (v). Since a terminal can be connected to only one central, such a condition is equivalent to the existence of a matching in BH covering CH . By Hall’s Theorem [16], such a matching exists if, and only if, condition (3) holds. P (ii) ⇔ (iii) Let H be a spanning tree of G(C), S ⊆ C and S¯ = C \ S. Subtracting v∈C dv to P both sides of the equality v∈C dH (v) = 2c − 2, we get v∈C

As the roles of S and S¯ can be interchanged above, we conclude that (ii) is equivalent to (iii). 

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Remark 1. If tree H stated in item (ii) (or (iii)) can be determined in polynomial time, a feasible solution of FMD-MST can also be found in polynomial time. As feasibility implies dv ≤ n, the construction of BH and so the determination of the matching to complete the minimum degree of the vertices v such that dH (v) < dv can be done in polynomial time. Every vertex in T not covered by such a matching is then connected to any adjacent central.

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As Theorem 2 suggests, the complexity of FMD-ST strongly depends on G(C). To focus on the structure induced by the centrals, we now assume that G has all possible edges between terminals and centrals, that is, the subgraph induced by the edges δ(T : C) is complete bipartite. We say in this case that G is T-C-complete. For T-C-complete graphs, Theorem 2 can be simplified as follows:

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Corollary 2. Suppose that G is T-C-complete. FMD-MST is feasible if, and only if, there exists P a spanning tree H of G(C) such that v∈C max{0, dH (v) − dv } ≤ t − tmin .

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Proof. The condition stated above is equivalent to item (iii) of Theorem 2. Indeed, since G is T-C-complete, |T (C \ S)| = t for all S ( C. In addition, for S = C, the inequality in item (iii) is always satisfied at equality (see (4)). 

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The above corollary shows that, if t = tmin , FMD-ST is equivalent to the decision version of DC-MST. As this problem is NP-Complete even in the d-uniform case, we have the following complexity result. Proposition 3. FMD-ST is NP-Complete, even in the d-uniform case and if G is T-C-complete. Although FMD-ST remains difficult when G is T-C-complete, it can become an easy problem if we additionally impose conditions on the number of terminals or on the structure of G(C). Examples of such conditions are given by the following proposition that complements Corollary 2. 8

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Proposition 4. Suppose G is T-C-complete. We can find a feasible solution of FMD-MST in polynomial time, if any of the following conditions holds: 1. t ≥ tmin + max{0, c − dmin − 1}, where dmin = minv∈C dv ; 2. t ≥ tmin + max{0, |C1 | − 2}, where C1 = {v ∈ C : dv = 1}, and G(C) has minimum degree at least dc/2e; 3. t ≥ tmin and dmin ≥ ∆∗ +1, where ∆∗ = min{∆(H) : H ∈ F} and F be the set of spanning trees of G(C).

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Proof. By Remark P 1 and Corollary 2, it suffices to find, in polynomial time, a spanning tree H of G(C) such that v∈C max{0, dH (v) − dv } ≤ t − tmin . In case 1, let H be any spanning tree of G(C). Since the maximum possible value of the summation in the lefthand P side would occur if H were a star centered at a vertex with degree requirement dmin , we have v∈C max{0, dH (v) − dv } ≤ max{0, c − 1 − dmin } ≤ t − tmin . In item 2, the result is trivial if c ≤ 2. When c > 2, Dirac’s Theorem ensures that G(C) is Hamiltonian and, in addition, a Hamiltonian path can be obtained in polynomial time between any pair a, b ∈ C [24]. In particular, take this pair such that P C1 = {a} (if |C1 | = 1) and P C1 ⊇ {a, b} (if |C1 | ≥ 2). Let H the Hamiltonian path. Thus, v∈C max{0, dH (v) − dv } = v∈C1 max{0, dH (v) − 1} = |C1 \ {a, b}| = max{0, |C1 | − 2} ≤ t − tmin . Item 3 follows from the fact that a spanning tree H of G(C) P with maximum degree ∗ ∆(H) ≤ ∆ +1 can be found in polynomial time [12]. Then, we have v∈C max{0, dH (v)−dv } ≤ P ∗  v∈C max{0, ∆ + 1 − dmin } = 0 ≤ t − tmin . 4. ILP Formulations

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In this section we present integer programming formulations for FMD-MST and theoretically compare the lower bounds provided by their linear relaxations. An experimental comparison of these formulations is presented in Section 6. As already observed, a feasible solution of FMD-MST consists of a spanning tree of G(C), and links between terminals and centrals so as to satisfy the minimum degree constraints. Thus, each proposed formulation apply MST constraints only to the subgraph G(C) induced by the centrals. Additional constraints ensure connectivity between terminals and centrals, as well as the minimum degree of each central. For simplicity, we assume that there is no edge between terminals, as those edges could not be part of a solution. Whenever the binary variable xe appears in a formulation, it indicates whether edge e ∈ E was chosen to enter the solution (xe = 1) or not (xe = 0). xC and xT are the vectors comprising the entries xe , for all e ∈ E(C) and all e ∈ δ(T : C), respectively. In general, if X ⊂ B|E(C)| is the set of incidence vectors of the spanning trees of G(C), FMD-MST can be described as: ( ) X (FMD) min ce xe : x ∈ X (5) e∈E

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(x, λ) ∈ B|E(C)| × Bc·|A(C)| λkijP+ λkji xe e∈E(C) P λkij

: = xe , ∀e = ij ∈ E(C) = c − 1, ≥ 1,

: = xe , =c−1

λkik

(j,i)∈δ − (i)

fji −

yir fij

(i,j)∈δ + (i)

yij ≤ fij

(x, y, u) ∈ B|E(C)| × B|A(C)| P yij + yji yij (i,j)∈δ − (j)∩A(C)

yir ui − uj + cyij 1 ≤ ui ur 10

    ∀k ∈ C, ∀e = ij ∈ E(C)     

∀(i, k) ∈ A(C)

= 0, = 1,

    ∀e = ij ∈ E(C)      ∀j ∈ C \ {r}   ∀(i, r) ∈ A(C) ∀i ∈ C\{r}

≤ (c − 1)yij , ∀(i, j) ∈ A(C) ×Rc : = xe , = 1,

(8)

      

= 0,

(i,j)∈δ − (j)∩A(C)

P

       

∀j ∈ C, ∀k ∈ C, k 6= j

(x, y, f ) ∈ B|E(C)| × B|A(C)| × R|A(C)| : P yij + yji = xe , yij = 1, P

∀S ⊂ C, r ∈ S

(7)

= 1,

(i,j)∈δ − (j)∩A(C)

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          

      

xe

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Xint = Projx

        

       

= c − 1,

e∈E(C)

X

   

xe

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  

X

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Xsec =

 x ∈ B|E(C)| :   

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There is a number of possibilities to describe X in terms of integer points of a polyhedron [14, 17, 18]. One direct strategy defines the polyhedron via the variables xe only. Another possibility is to transform the input undirected graph G(C) = (C, E(C)) into a directed graph D = (C, A(C)) and to consider variables related to the arcs of D too. The digraph D has arcs (i, j) and (j, i) for every edge e = ij ∈ E(C), both with cost ce , and an arbitrary node r ∈ C is chosen to be the root. A tree in G(C) corresponds to an arborescence in D rooted at r, and vice-versa. In this work, we consider the following alternative descriptions of X, where Projx (W ) stands for the set {x : (x, y) ∈ W, for some y}.

∀e = ij ∈ E ∀j ∈ C\{r}

= 0, ∀(i, r) ∈ A(C) ≤ c − 1, ∀(i, j) ∈ A(C), j 6= r ≤ c − 1, ∀i ∈ C\{r} =0

         

                    

       

(9)

(10)

(11)

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All these formulations are based on the fact that a spanning tree is a subgraph that satisfies two of the following conditions: (i) its number of edges is one less than its number of vertices; (ii) it is connected; (iii) it has no cycles. Xsec avoids cycles by the subtour elimination constraints r (SEC) [9]. Xcut ensures connectivity by the directed cutset constraints (DCUT) [17]. The idea behind Xint is to define c arborescences, each one rooted at a different vertex k ∈ C but all of them using the same underlying edges. It can be shown that this “intersection” at the same edges guarantees connectivity, without the need of the cut constraints [18]. On the other hand, the r connectivity in Xflo is obtained by sending c − 1 units of flow from r, each unit being addressed r to a different vertex in C \ {r} [17]. Finally, Xmtz avoids cycles by labeling the vertices in such a way that the label of j is strictly greater that the label of i if arc (i, j) is chosen [14]. This is guaranteed by the Miller-Tucker-Zemlin subtour elimination constraints [22]. More details on these formulations for the MST problem can be found in the cited references. In general, any formulation for MST could be a basis to define a formulation for FMD-MST as in (5)-(6). Among the several options, we choose (7)-(11) to carry out a comparison for the following reasons:

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• strength of the relaxation: As formulations (5)-(6) essentially differ from one another by the description of X, it may be expected that stronger formulations for MST would r provide stronger formulations for FMD-MST. This justifies the choice of Xsec , Xint and Xcut , whose linear relaxations have integral extreme points.

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• size of the model: The number of variables and constraints certainly are parameters that affect a formulation performance. Although with potential to provide stronger bounds, r Xsec and Xcut have O(c2 ) variables and O(2c ) constraints, which may bring about a high computational effort (even the separation problem being polynomial). On the other hand, r r seem to provide weaker formulations but have only O(c2 ) variables and and Xmtz Xflo constraints. As an intermediate option, Xint has the integrality property and uses O(c3 ) variables and constraints.

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• performance with related problems: Experiences with related problems should not be r neglected. Xsec and Xcut have been commonly used to model DC-MST [4, 5, 7] whereas good r r and Xint [2, 3, 21]. , Xmtz computational results have been reported to MD-MST with Xflo 4.1. Theoretical Comparison of the Linear Relaxations

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Each of the presented descriptions for X leads to a feasible set X and a corresponding ILP formulation (FMD). The same super- and sub-indices used to identify each description of X will be r r used to identify the corresponding X and formulation. For instance, X = Xcut yields Xcut and r ¯ FMDcut . In addition, let X stand for the polyhedron obtained when the integrality constraints are discarded in the definition of X. Similarly, let X¯ be the polyhedron given by (6) with x ∈ Bm , ¯ In other words, X¯ defines the linear relaxation of FMD. xC ∈ X replaced by x ≥ 0, xC ∈ X. r r ¯ int ⊆ X ¯ r and conv(X) ⊆ X ¯ mtz ¯ sec = X ¯ cut , =X It is known that, for every r ∈ C, conv(X) = X flo where each inclusion can be strict [17, 18, 2, 14]. This implies that the linear relaxations of the r ILP formulations for MST given by Xsec , Xcut or Xint provide the optimal value. The same does r r not necessarily occur with the formulations based on Xflo and Xmtz , where the choice of root r ∈ C can affect the quality of the lower bound. The relations stated above lead to a direct comparison among the linear relaxations of FMD-MST. r r Proposition 5. For all r ∈ C, we have that X¯sec = X¯cut = X¯int ⊆ X¯flo and X¯sec ⊆ X¯mtz .

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c1

0

c2

t1

M 0

c1

0

0

c3

0

c4

t1

0

0.5 c3

t2

0.5 0.5

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(a) Instance of FMD-MST.

c2

0.5

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By Proposition 5, the lower bound provided by FMDrcut is greater than or equal to the bounds provided by the other formulations. In what follows we show through an example that even this best bound can be far from the value of an optimal integer solution. Consider the graph in Figure 4(a), where the edges have cost 0 or M , as indicated. Let c1 , c2 , c3 , c4 be the centrals and t1 , t2 be the terminals. Consider d = 2 and r = c1 . It is easy to realize that the two edges of cost M should be in any feasible solution, due to the degree constraints. Thus an optimal solution has cost 2M . On the other hand, the optimal solution of the linear relaxation of FMDrcut is shown in Figure 4(b). The cost of this solution is M , so the integrality gap is M . By the symmetry of the graph, note that this result is independent of the choice of the root.

0.5

0.5

c4

(b) Solution of the linear relaxation of FMDrcut .

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Figure 4: Integrality gap of size M .

Although the above example shows that the bound provided by the linear relaxation of any of the formulations may be far from the optimum, the presented scenario is unlikely to happen in real applications or random instances. Besides, in the example, fixing one y variable associated with edges of cost M leads to infeasibility (if fixed to 0) or to the optimal solution (if fixed at 1). This suggests that the lower bound can greatly increase during an enumeration process.

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4.2. Practical Issues Some aspects become relevant when attempting to solve the presented ILP formulations. To code the formulations that depends on the Projx operator, we keep the variables (besides x) used in the definition of X. In other words, formulation (5) is expressed in terms of x and the other variables (e.g, y in FMDrcut ). Actually, the relations xe = yij + yji can be used in FMDrcut , FMDrflo and FMDrmtz to eliminate the variables x. All expressions, including the objective function and the degree constraints, replace the variables x by the variables y appropriately. Similarly, we can discard x in FMDint . This reduces the size of these models. In the case of FMDsec and FMDrcut , the complete formulation is computationally intractable for moderately sized instances, due to the exponential number of constraints. However, SECs and DCUTs can be separated in polynomial time [25, 17]. The in-degree constraint X yij = 1, ∀j ∈ C \ {r} (12) (i,j)∈δ − (j)∩A(C)

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r (that appears in several formulations) are redundant in Xcut . They are implied by the constraint on the number of edges together with the DCUTs defined by S = C \ {j}, for all j ∈ C \ {r}. However, they are added to the relaxation to accelerate the separation process. For each choice of the root r, we obtain a different formulation FMDrcut , FMDrflo and FMDrmtz . Although this choice may affect the performance of the formulation (for example, lower bounds depend on r in FMDrflo and FMDrmtz ), we will not deeply study this issue. The MST formulations not having the integrality property could be strengthened. Particularly, as suggested by Gouveia [14], the MTZ inequalities

ui − uj + cyij ≤ c − 1, ∀(i, j) ∈ A(C), i, j 6= r, 12

(13)

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can be lifted to X

ykj + ui − uj + cyij + (c − 3)yji ≤ c − 1, ∀(i, j) ∈ A(C), i, j 6= r.

(k,j)∈δ − (j)\(i,j)

Using (12), we have that the summation in the above inequality is equal to 1 − yij so that it can be rewritten as ui − uj + (c − 1)yij + (c − 3)yji ≤ c − 2, ∀(i, j) ∈ A(C), i, j 6= r,

(14)

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We remark that (14) dominates the following valid inequality suggested by Desroches and Laporte [10]: ui − uj + cyij + (c − 2)yji ≤ c − 1, ∀(i, j) ∈ A(C), i, j 6= r, (15) provided that yij + yji ≤ 1. We tested the formulation using alternatively (13), (14) or (15).

5. Lagrangian Relaxation

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Removing the degree constraints of the centrals yields a polynomial problem. It is equivalent to the 1-uniform case (see Proposition 1). Instead of simply removing such constraints, we can penalize them in the objective function via Lagrangian multipliers. The resulting Lagrangian dual problem is the basis for obtaining lower and upper bounds.

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5.1. Lagrangian Dual Consider Pthe formulation based on subtour elimination constraints (FMDsec ). Dualizing every constraint e∈δ(i) xe ≥ di with a multiplier ui ≥ 0, ∀i ∈ C, we get the following Lagrangian function: X X X X X ψ(x, u) = ce x e + ui (di − xe ) = c¯e xe + ui di , e∈E



e∈δ(i)

e∈E

i∈C

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where, for every e = ij ∈ E,

i∈C

c¯e =

ce − ui − uj , if (i, j) ∈ C × C, ce − ui , if (i, j) ∈ C × T.

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Thus, every u ≥ 0 provides the lower bound     X L(u) = min ψ(x, u) | (xC , xT ) ∈ Xsec × B|δ(C:T )| , xe = 1, ∀i ∈ T . x  

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The feasible points of the above program are incidence vectors of spanning trees of G where the vertices in T are leaves. By Proposition 1, it can be polynomially solved. It is a matter of determining a minimum spanning tree of G(C) with costs c¯e , ∀e ∈ E(C), and then assigning each terminal j to a central i? ∈ arg min{¯ cij : ij ∈ δ(C : T )}. The best of these lower bounds is given by the Lagrangian dual (LD)

L(u∗ ) = max L(u) u≥0

¯ , to It is know that L( ) is a piecewise linear concave function [29]. P One subgradient of L( ) at u be denoted g(¯ u), is such that its i-th component is gi = di − e∈δ(i) x ¯e , where x ¯ is a solution of ¯ ). L(¯ u), that is, L(¯ u) = ψ(¯ x, u 13

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A classical iterative method to solve (LD) is the Subgradient Method (SM). Roughly speaking, the iterate uk+1 is a move from uk in the direction of g(uk ). The Deflected Subgradient Method (DSM) is a variant of SM, where the current moving direction is a convex combination of the subgradient and the previous moving direction [6, 8]. Precisely, starting from u0 , we define uk+1 for any k ≥ 0 such that uk+1 = max{0, uki + θk ski }, i where

sk = (1 − λk )g(uk ) + λk sk−1 θk = πk (U B − LB)/ksk k22 ,

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is the (deflected) moving direction, for some chosen λk ∈ [0, 1), and

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is the stepsize, defined in terms of a decreasing parameter πk > 0 and the best known lower and upper bounds (LB and UB, respectively). Initially, we set λ0 = 0. Martinez & Cunha reported good results of DSM applied to MD-MST [20]. Motivated by this experience, we now apply the method to FMD-MST. We start with u0 = (1/c)1 and use strategies similar to those in [20] to set the parameters. Initialized at 2, parameter π is halved after N = 50 iterations without increasing the best known lower bound, i.e. after N iterations with L(uk ) < LB. Parameter λ is kept fixed at 0.05. The iterative process stops if one of the following criteria holds: (i) πk ≤ πmin := 0.1, (ii) U B − LB ≤ 0.9, or (iii) the time limit is exceeded. See Algorithm 1, where we have already introduced calls to a Lagrangian heuristic to be described in the following. It converts the solution of L(uk ) into a feasible tree.

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Algorithm 1: Lagrangian Algorithm Input: An upper bound U B, a lower bound LB (possibly trivial values); Output: An upper bound U B, a lower bound LB (possibly improved bounds); 1 k ←0 ; /* Iteration counter */ 0 2 u = (1/c)1; nstuck ← 0; π = 2; 3 while time limit not exceeded do 4 Solve L(uk ) and find a (possibly infeasible) tree H; 5 LB ← max{LB, L(uk )}; 6 if LB = L(uk ) then nstuck ← 0 else nstuck ← nstuck + 1; 7 Apply LagrangianHeuristic(H); 8 U B ← min{U B, c(H)}; 9 if U B − LB ≤ 0.9 then Break (go to line 18); 10 Determine uk+1 using λk = 0.05 and πk = π; 11 if nstuck = 50 then 12 nstuck ← 0; π ← π/2; 13 if π ≤ 0.1 then Break (go to line 18); 14 end if 15 k ← k + 1; 16 end while 17 if time limit not exceeded then 18 Apply LagrangianHeuristic(H) ; /* Last Lagrangian Iteration */ 19 U B ← min{U B, c(H)}; 20 end if 21 return U B, LB;

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5.2. Feasibility Procedures At each iteration of the DSM, we obtain a spanning tree H of G that satisfies the unit degree of the terminals but usually violates the minimum degree constraints of some centrals (Step 4 of Algorithm 1). We can attempt to convert such an infeasible solution into a feasible one by interchanging edges from E(H) with edges from E \ E(H). Ideally, we want these changes to increase the least possible the cost of the new tree. Note that a spanning tree H of G with dH (v) = 1 for all v ∈ T is a feasible solution of the 1-uniform FMD-MST. For easy reference, such a tree will be called 1-uniform tree. An edge-swap operation with respect to an 1-uniform tree H is defined by a pair of edges (e, e0 ), e ∈ E(H), e0 ∈ E \ E(H), such that (H \ {e}) ∪ {e0 } is a tree. Given an edge-cost function w : E → R, the cost of an edge-swap (e, e0 ) is w(e0 ) − w(e). For each i ∈ C such that dH (i) < di , we define the following edge-swaps that increase the degree of i: • edge-swap iT : the minimum-cost edge-swap in the set {(e, e0 ) : e0 = (i, `), e = (j, `), ` ∈ T, j ∈ C, dH (j) > dj }. • edge-swap iC : the minimum-cost edge-swap in the set {(e, e0 ) : e0 = (i, `), e = (j, `), ` ∈ C, j ∈ C, dH (j) > dj , e in the cycle of H ∪ {e0 }}.

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• edge-swap i< : the minimum-cost edge-swap between iT and iC .

We call Feasibility Procedure (FP) any procedure that, given an 1-uniform tree H and an edge function w, iteratively applies one of the edge-swaps iT , iC and i< , for any i ∈ C such that dH (i) < di , thus getting a new tree H, while this operation is possible. See Algorithm 2. We show that this general procedure indeed yields a feasible solution, if it exists, when G is complete.

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Algorithm 2: Feasibility Procedure Input: A 1-uniform tree H, an edge-cost function w Output: A feasible tree H < 1 C ← {i ∈ C : dH (i) < di }; < 2 while C 6= ∅ and edge-swap can be applied do 3 Let i ∈ C < ; 4 Apply iT , iC or i< to obtain new tree H; 5 if dH (i) = di then C < ← C < \ {i}; 6 end while 7 return H;

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Proposition 6. Suppose that G is a complete graph and t ≥ tmin . Starting from any 1-uniform tree and any edge function, a Feasibility Procedure ends up with a feasible solution of FMD-MST.

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Proof. Let H be an 1-uniform tree and C < = {i ∈ C : dH (i) < di }. If C < = ∅, H is a feasible solution and is the output of a Feasibility Procedure. Otherwise, let i ∈ C < . We claim that there exists j ∈ C such that dH (j) > dj . If not, dH (i) < di , dH (j) ≤ dj , ∀j ∈ C, and dH (j) = 1, ∀j ∈ T , which imply X X 2(t + c − 1) = dH (j) < dj + t, P

j∈V

j∈C

that is, t < j∈C dj − 2c + 2 = tmin . Thus, we get a contradiction. Given such a j, we have 2 cases. If there is ` ∈ T such that e = (j, `) ∈ H, we can apply iT . Indeed, e0 = (i, `) ∈ E \ E(H) because G is complete and dH (`) = 1. In the complementary 15

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case, since dH (j) > 1, there must be at least two centrals that are neighbors of j in H. Of course, the unique path in H from i to j does not pass by one of them, say ` ∈ C. Moreover, e0 = (i, `) ∈ E \ E(H), since H is acyclic, and e = (j, `) ∈ H belongs to the cycle formed in H ∪ {e0 }. Therefore, we can apply iC . In any case, an edge-swap can be applied to decrease the total degree infeasibility by 1. As FMD-MST is feasible when G is complete and t ≥ tmin , a FP ends up with a feasible solution. 

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Observe that Algorithm 2 may lead to different results according to the implementation of steps 3-4. We specialized it into 4 procedures FP1, FP2, FP3 and FP4. They differ from one another in the choice of i in Step 3. For the chosen i, all of them apply i< . In FP1, i is always the vertex with smallest index in C < . In other words, an arbitrary fixed order is followed to choose i (the order that the vertices appear in the input file was used). In FP2, i is the vertex in C < such that i< has the smallest cost. In FP3, i is chosen at random in C < . FP4 is a slight modification of FP3: once selected, i is kept until its minimum degree is attained.

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5.3. Improvement Procedures Once we have a feasible solution of FMD-MST, we can try to improve it in several ways. For instance, we can keep the chosen edges between centrals and look for the best connections between centrals and terminals. This can be done by solving a flow problem, as the next proposition shows. Proposition 7. Let F be a spanning tree of G(C). The problem of determining the best feasible solution H of FMD-MST such that H(C) = F is polynomial.

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Proof. We show that such a problem reduces to a minimum-cost flow problem in the network illustrated in Figure 5. The nodes of the network are all the terminals and centrals, and two additional nodes s and g. For edge ij ∈ E(G) with i ∈ T and j ∈ C, we create an arc with capacity 1 and cost cij . We create an arc from s to each terminal i with capacity 1 and cost 0. We also create an arc from each central j to g with cost 0 and infinite capacity. The supply of node s is t = |T |. Each terminal node has demand 0, and each central j has demand P δj = max{0, dj − dF (j)}. The demand of node g is t − j∈C δj .

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T

(1, 0)

0 i

(1, cij )

δj j

(∞,0) t −

P g

j∈C

δj

PT

t s

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Figure 5: Network Illustration. Node s is the unique surplus node.

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Let H be a feasible solution of FMD-MST such that H(C) = F . Then δH (T : C) induces a feasible flow. It sends one unit of flow by each arc (s, i), i ∈ T , which follows through the arc (i, j), where j ∈ C is defined by the edge ij ∈ δH (T : C). Since H(C) = F , the total amount of flow reaching each j ∈ C is at least δj . The excess can always go to g. On the other hand, each feasible flow induces an assignment of the terminals to the centrals. Since at least δj terminals are assigned to central j, for every j ∈ C, this assignment together with F leads to a feasible solution of FMD-MST. As the transformations in both directions keep the cost, the equivalence between the two problems follows.  Proposition 7 suggests a procedure to improve a feasible solution H of FMD-MST: keep the spanning tree H(C) of G(C) and re-allocate the terminals by solving a minimum-cost flow problem (starting the flow algorithm with the feasible flow defined by δH (T : C)). This procedure will 16

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Algorithm 3: Local Search Input: A feasible tree H, an ordered list Q containing a subset of E(H); Output: A feasible tree H with smaller or equal cost; 1 N ← ∅; /* List of edges added to H */ 2 foreach edge e = (i, j) in the order of list Q do 3 foreach edge e0 = (i0 , j 0 ) after edge e in list Q do 4 S1 ← {(i, j 0 ), (i0 , j)}, S2 ← {(i, i0 ), (j, j 0 )}; 5 if there is S ∈ {S1 , S2 } satisfying conditions (C1), (C2), and (C3) then 6 H ← (H \ {e, e0 }) ∪ S; 7 Q ← Q \ {e, e0 } ; /* the ’for’ in line 2 skips e0 */ 8 N ← N ∪ S; 9 Break (go to line 2); 10 end if 11 end foreach 12 end foreach 13 while N 6= ∅ do 14 Let e = (i, j) ∈ N ; 15 N ← N \ {e}; 16 foreach e0 = (i0 , j 0 ) in the order of list Q do 17 S1 ← {(i, j 0 ), (i0 , j)}, S2 ← {(i, i0 ), (j, j 0 )}; 18 if there is S ∈ {S1 , S2 } satisfing conditions (C1), (C2), and (C3) then 19 H ← (H \ {e, e0 }) ∪ S; 20 Q ← Q \ {e0 }; 21 N ← N ∪ S; 22 Break (go to line 13); 23 end if 24 end foreach /* It reaches this line only if e is not replaced */ 25 Q ← Q ∪ {e}; 26 end while 27 return H;

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be denoted MINCOST-DEGREE-COMPLETION. Although polynomial, it can be very time-consuming if applied several times. In addition, it only considers changes in edges involving terminals. An alternative procedure is to elect a subset Q of edges of the incumbent tree H as candidate edges to be replaced. In this vein, we propose the 2-edge-swap procedure described in Algorithm 3. It performs a local search. Basically, we replace two non-adjacent edges e = (i, j) ∈ Q and e0 = (i0 , j 0 ) ∈ Q by the edges f = (i, j 0 ) and f 0 = (i0 , j), if the following three conditions hold: (C1) f and f 0 do exist in G; (C2) cf + cf 0 < ce + ce0 ; (C3) (H \ {e, e0 }) ∪ {f, f 0 } is also a tree. Since we are not altering the degrees of the vertices, these conditions guarantee that the modified tree is an improved feasible solution. Observe that H \ {e, e0 } has three connected components, and each of them contains (at least) one vertex in U = {i, j, i0 , j 0 }. Assuming that conditions (C1)-(C2) hold (which is easily checked), condition (C3) is satisfied if, and only if, the two (necessarily distinct) nodes from U in the same connected component are i, i0 or j, j 0 . This is particularly easy to check whenever e, e0 ∈ δH (C : T ). In this case, the required nodes are exactly the two centrals in U .

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Algorithm 3 has two main phases. The first one runs from Step 2 to Step 12. It takes the first edge e from Q (which is implemented as a linked list) and attempts to combine it with any other subsequent edge e0 from Q. As soon as a 2-edge-swap improves the solution, it is updated; e and e0 are removed from Q whereas the 2 new edges are put in another linked list N . The process is repeated with every remaining edge in Q. The second phase (steps 13-26) attempts to combine the new edges (stored in N ) with other edges (kept in Q in the first phase or moved from N to Q in the second phase). Once an edge e in N is taken, it iteratively tries an improving 2-edge-swap with an edge e0 from Q. If possible, e is removed from N , e0 is removed from Q and the 2 new edges are added to N . Otherwise, e is moved to Q. The process stops when N becomes empty. Note that Algorithm 3 converges. The first phase stops because no element is added to Q during its execution. The second phase also comes to an end. Indeed, Q does not increase during the inner loop. Also, at each major iteration, N always loses an element and only gains a new one if a strictly better tree is found. 5.4. Lagrangian Heuristics

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At each iteration k of the DSM, we obtain a 1-uniform tree H given by the solution of L(uk ). We propose three heuristics to convert H into a feasible solution. We use the feasibility procedures and the improvement procedures described in subsections 5.2 and 5.3. Along this subsection, we assume that G is complete and t ≥ tmin so as to guarantee such a conversion by Proposition 6. If G is not complete, we can add the missing edges with very high cost, as already observed. However, in this case, we recall that the feasibility problem is hard (Proposition 3). The first heuristic, LH1, initially applies the feasibility procedure FP1 starting from H and the edge-cost function w = c (the costs that define the instance). The resulting feasible tree H is then submitted to Algorithm 3 with Q = δH (C : T ). Actually, at the last iteration of the Lagrangian algorithm (if it converges within the time limit), LH1 proceeds slightly different. Algorithm 3 is applied with Q = δH (C) (instead of Q = δH (C : T )) and then the resulting tree is improved by MINCOST-DEGREE-COMPLETION. This use of the local search algorithm provided better trade-off between quality of the solution and processing time than other strategies that we have tried. See Algorithm 4.

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Algorithm 4: Langrangian Heuristic LH1 Input: A 1-uniform tree H Output: A feasible tree H 1 if last Lagrangian iteration then 2 Apply Local Search to H and δH (C); 3 Apply MINCOST-DEGREE-COMPLETION to H; 4 else 5 Apply Feasibility Procedure FP1 to H and c; 6 Apply Local Search to H and δH (C : T ); 7 end if 8 return H; The second heuristic, LH2, is similar to LH1 except for the edge-cost function used in the feasibility procedure (Step 1 of Algorithm 4). Now, instead of c, w is given by the Lagrangian costs c¯k , where c¯ke = ce − uki − ukj , if e = (i, j) ∈ C × C, and c¯ke = ce − uki , if (i, j) ∈ C × T . The third heuristic, LH3, uses a modified Kruskal’s Algorithm to obtain a feasible solution, based on the current Lagrangian costs c¯k . See Algorithm 5. The procedure has two main phases. In the first one (` ≤ p), it determines a spanning tree of G(C) that can be extended to a feasible 18

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solution. This extension is the aim of the second phase (` > p). After finding a feasible solution, it is improved by Algorithm 3 and MINCOST-DEGREE-COMPLETION, exactly as in LH1.

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Algorithm 5: Langrangian Heuristic LH3 Input: A tree H Output: A feasible tree H 1 if last Lagrangian iteration then 2 Apply Local Search to H and δH (C); 3 Apply MINCOST-DEGREE-COMPLETION to H; 4 else 5 Let e1 , . . . , ep , p = |E(C)|, and ep+1 , . . . , ep+q , q = |δ(C : T )|, be the edges in E(C) and δ(C : T ), respectively, ordered according to the nondecreasing values of c¯ke , e ∈ E; 6 H ←P (C ∪ T ; ∅); P 7 K = i∈C max{0, di − dH (i)}; M = 2(c − 1) − i∈C dH (i) + t; 8 for ` = 1, . . . , p, p + 1, . . . p + q do 9 if ` ≤ p then r ← 2 else r ← 1; 10 Let ke = |{i ∈ e : dH (i) < di }| ; /* di = 1 if i ∈ T */ 11 if H ∪ e is acyclic and M − r ≥ K − ke then 12 H ← H ∪ e; 13 M = M − r; K = K − ke ; 14 end if 15 end for 16 Apply Local Search to H and δH (C : T ) 17 end if 18 return H;

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The first phase of LH3 starts by ordering the edges in E(C) according to the nondecreasing order of the Lagrangian costs. Following this order, each edge is considered to be included in the tree. At a given iteration, let HPbe the forest formed so far and let e = ij ∈ E(C) P\ E(H) be the incumbent edge. Let K(H) = i∈C max{0, di − dH (i)} and M (H) = 2(c − 1) − i∈C dH (i) + t. Still, let ke ∈ {0, 1, 2} be the number of endpoints of e whose degree in H is less than its required minimum degree. Edge e is included in H if H ∪ {e} is acyclic and M (H) − 2 ≥ K(H) − ke . If G is complete, the second condition guarantees that it is still possible to satisfy the minimum degree conditions. Indeed, K(H) is the minimum number of edges (to be added to H) necessary to attain the minimum degrees. On the other hand, M (H) is the remaining contribution to the degreesP of the centrals that can be given by the edges (to be added to H) between centrals (2(c − 1) − i∈C dH (i)) and between centrals and terminals (t). Since M (H ∪ {e}) = M (H) − 2 and K(H ∪ {e}) = K(H) − ke , the stated condition has the claimed rationale. The second phase proceeds in a similar manner. The edges in δ(C : T ) are considered sequentially, according to the nondecreasing order of the Lagrangian costs. An edge e is included in H if H ∪ {e} is acyclic and M (H) − 1 ≥ K(H) − ke . Now, M (H ∪ {e}) = M (H) − 1 and K(H ∪ {e}) = K(H) − ke , where ke ∈ {0, 1}. Notice that, if e = ij, i ∈ T , is included in H, then no other edge involving i can enter H. Observe that LH3 uses the same ordering of the edges as the Kruskal’s algorithm applied to solve L(uk ). Thus, its (additional) computational cost is to keep M () and K() while searching this edge list. It is worth remarking that we have tried several combinations of the feasibility and improvement procedures. For instance, in LH1 and LH2, we have replaced FP1 by either FP2, FP3, 19

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or FP4. On the other hand, we have tried to apply MINCOST-DEGREE-COMPLETION and/or Algorithm 3 with δH (C) in every iteration of the Lagrangian algorithm. These strategies were not able to yield better computational results in the time limit we adopted. 6. Computational Results

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We have implemented the ILP formulations and algorithms described in the previous sections. Their computational performance was evaluated using randomly generated test instances. The main computational results are reported in this section. Complementary details and results are given in Appendix A and https://github.com/fabiocsd/FMD-MST. The algorithms were implemented in C/C++ language and ran on an Intel Xeon(R), with 3.30 GHz and 8 GBytes of RAM memory, using Linux operating system. CPLEX 12.6.1 was used to solve the LP and ILP programs. Each run with a given instance was limited to 18000 seconds (5 hours) of processing time. Since more than one processor may be running simultaneously, observe that the limit for the elapsed time is usually much smaller.

Table 1: ALM test instances c 20, 30, 40, 49 40, 50, 60, 70, 80, 90, 99 60, 70, 80, 90, 100, 110, 120, 130, 140, 149 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 199 60, 70, 80, 90, 100, 120, 140, 160, 180, 200, 220, 240, 249

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6.1. Test Instances In our computational experiments, we used three sets of instances. The first one is adapted from test instances for the problem MD-MST, denoted ALM [19]. These instances are shown to be easily solvable by some of the ILP formulations we presented in Section 4, as will be seen in the following. So, we randomly generated two other sets of instances, to be denoted EUC and NEU. The original ALM instances were generated by Martins & Souza [19]. Each of them is a complete graph with 100, 200, 300, 400, or 500 vertices. The vertices correspond to randomly generated points (according to a uniform distribution) in the Euclidean plane within a rectangle of dimensions 480×640. The cost of an edge is the Euclidean distance between the corresponding points. In each instance, the minimum degree requirement is a fixed value d for all vertices. From these ALM instances of MD-MST, we obtained d-uniform instances of FMD-MST. We just needed to partition the n vertices into c centrals and n − c terminals. For simplicity, we named central the first c vertices in the original instance, where the chosen values for c are presented in Table 1. Assuming d = 3, these values respect the bound c ≤ (n − 2)/(d − 1) (or equivalently t ≥ tmin ) and so guarantee feasibility whenever G is complete (see Proposition 4(2)). Since the graph is complete, choosing a different set of c centrals does not significantly affect the computational results.

# instances 12 21 30 45 39

For a given value of n, there are 3 original ALM instances. So, the numbers of centrals proposed in Table 1 lead to 147 distinct instances for FMD-MST with d = 3. Actually, we only present computational results for the ALM instances with d = 3. Larger values of d restrict c to be smaller (c ≤ (n − 2)/(d − 1)). By its turn, smaller values of c tend to produce easier instances. In order to generate the other two sets EUC and NEU, we considered three aspects that affect the hardness of an instance: (i) the relation between n and c, (ii) the distribution of the 20

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HHg H 60 100 200 300 400 500 600 700 800 900 k

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Table 2: Number of vertices n = kc + 2. EUC Instances NEU Instances

116 192 382 572 762 952 1142 1332 1522 1712 1.9

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minimum degree values di , i ∈ C, and (iii) the edge cost distribution. We have used the following strategies. For EUC, the graphs and costs are generated in the same manner as in ALM, so that the edge costs satisfy the triangular inequality. For NEU, each graph is also complete but the edge costs are uniformly distributed values in [1, 1000]. The vertex partition into centrals and terminals as well as the minimum degree requirements follow the same strategy in EUC and NEU, but it differs from that one used in ALM. First, we choose the number c of centrals of the instance from the set {60, 100, 200, 300, 400, 500, 600, 700, 800, 900}. For a given c, we generate 4 instances of each set, to be denoted EUCc-g or NEUc-g, for g = 1, 2, 3, 4. Thus, each set has 40 instances. P We have noticed that the problem becomes harder when the inequality n ≥ i∈C di − c + 2 (or equivalently t ≥ tmin ) tends to an equality. P P In the instances marked with g = 1, 2, we defined n = i∈C di − c + 2 + 0.4c whereas n = i∈C di − c + 2 when g = 3, 4. See Table 2. In other words, for each c, the instances EUCc-g and NEUc-g have exactly the minimum number of terminals required for feasibility (if g = 3, 4) or a “slack” of 0.4c terminals (if g = 1, 2).

146 242 482 722 962 1202 1442 1682 1922 2162 2.4

182 302 602 902 1202 1502 1802 2102 2402 2702 3

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The difference between the groups identified with g = 1 and g = 2 (as well as g = 3 and g = 4) lies in the distribution of di , i ∈ C. For g = 1, 3, we adopted di = d = 3, ∀i ∈ C, that is, instances from these two groups (as well as the ALM instances) are 3-uniform. For EUC and g =P2, 4, we used di = 2, 3, 4 for 25%, 60% and 15% of the centrals, respectively. This leads to i∈C di = 2.9c. For NEU and P g = 2, 4, we have di =P3, 4, 5 for 60%, 20% and 20% of the centrals, respectively. In this case, i∈C di = 3.6c. Since i∈C di is a multiple of c in each case, we have that n = kc + 2, for some k (see Table 2). For easy reference, the instances in ALM and EUC are called Euclidean, as the costs are defined by Euclidean distances. The instances in NEU are named non-Euclidean. For DC-MST, it is observed that non-Euclidean instances are harder to solve (see [4] and references therein). 6.2. ILP Formulations

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Both procedures can identify up to c − 1 violated constraints (non-necessarily distinct), each one associated with a vertex in C \ {r}. This gives rise to different cutting-plane procedures. We tested the following strategies to add cuts at each iteration: (i) only the first encountered violated constraint; (ii) all encountered violated constraints; (iii) all different violated constraints found. Additionally, we combined them with the possibility or not of removing previously added constraints that become inactive. Here, we present the computational results only for the most effective combinations. Concerning the linear relaxations, the best performance for SECs was obtained by strategy (ii), whereas for DCUTs the best performance was due to strategy (iii) combined with the removal of previously added cuts whose slack is greater that  = 0.5 (other values for  were also tested). After solving the linear relaxation, the variables are turned into integers, and the same cuttingplane procedure is applied with a slight modification: inactive constraints are no more removed ( = +∞ in this second phase). The next three tables compare the formulations. For each formulation and instance, we B−LP , where LP is the optimal value of the calculate the percentage gap as %GAP = 100 BUBLB linear relaxation, BU B and BLB are the best upper and lower bounds obtained in the time limit, respectively. If the optimal value (OP T ) is known, we have BU B = BLB = OP T . This is the case of all ALM instances (Table 3), all EUC and NEU instances with c ≤ 300 (Table 4), 4 EUC instances and 14 NEU instances with c ≥ 400 (Table 5). The values of BU B and BLB and how they are obtained will be indicated in tables A.9, A.10, and A.11. The same root r was chosen in all directed formulations to be the vertex of smallest index. We observed that other choices lead to similar conclusions about the computational results.

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Table 3: ALM Instances - Results for the ILP formulations. Linear Relaxation Integer Formulation # solved %GAP Time (s) # solved Time (s) Form. instances Avg. Std. Dev. Avg.(∗) instances Avg.(∗) FMDrcut 147 0.022 0.060 234.35 147 259.32 FMDsec 147 0.022 0.060 166.89 146 170.34 FMDrflo 147 7.070 2.229 25.37 129 2997.54 FMDrmtz (14) 147 3.287 1.320 1.54 55 2539.59 FMDrmtz (15) 147 3.287 1.320 1.69 50 2322.98 FMDint 114 0.022 0.060 3506.68 93 2092.83 (*) Considering the solved instances only.

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Table 3 presents the average and standard deviation of the percentage gaps for the 147 ALM instances. It also shows the average processing times for the relaxed and integer formulations. When analysing the average times, one should have in mind that only solved instances are considered. For these test instances, the lower bounds provided by FMDrcut and FMDsec are very tight. Actually, most of the relaxed solutions are already integral (in 96 out of 147 instances). Even the optimal non-integral solution values are very close to the optimum (the average gap for these 51 instances is 0.06%). The very small integrality gaps led FMDsec and FMDrcut to optimally solve the ALM instances within the time limit (except for FMDsec in one instance with n = 500, c = 249). Actually, FMDrcut took 5500s to solve the hardest instance. We noticed, however, that the processing time increases exponentially with c. By Proposition 5, we know that FMDint provides the same gap as FMDsec and FMDrcut . However, the linear relaxation of this formulation could be solved only for 114 (out of 147) ALM instances, all of them with c ≤ 150. Moreover, the processing times were much higher. In the 33 remaining instances, the required memory was superior to the available memory. The performance of the integer formulation was also inferior: fewer solved instances and higher spent time. Regarding formulations FMDrflo and FMDrmtz , none of the (easily) obtained relaxed solutions 22

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is integral. The lower bounds provided by FMDrflo are generally worse than those provided by FMDrmtz (either with liftings (14) or (15)), which are significantly worse than those given by the other three formulations. Using the original MTZ constraints (13) instead makes the average percentage gap grow from 3.287 to 8.039, while the average processing times decreases to 0.43s. These formulations could optimally solve only part of the ALM instances. We have also implemented a multi-commodity flow-based formulation for FMD-MST. It could find relaxed solutions for 95 ALM instances and optimal solutions for only 86 ALM instances, taking 2342.74s and 2201.41s on average, respectively. As for MD-MST [2], it was outperformed by the single flow-based formulation. Similar experiments were carried out with the EUC and NEU instances. Due to their even inferior performances, we do not present computational results for FMDint and the multi-commodity flow-based formulation. For c ≤ 300, essentially the same conclusions can be inferred. See Table 4. For c ≥ 400, one main difference is that the linear relaxations of FMDsec , FMDint , and FMDrcut could not be solved within the time limit and memory available for all EUC and NEU instances (except for EUC400-1 and EUC-400-2 with FMDsec , EUC400-3 and EUC400-4 with FMDrcut ). Observe, however, that the incumbent solution value during the cutting-plane process used to solve FMDrcut (as well as FMDsec ) is also a lower bound. Table 5 presents the %GAP given by the lower bound obtained in the last iteration before stopping the cutting-plane algorithm. The analysis of the linear relaxations is now refined by groups.

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Table 4: EUC and NEU Instances with c ≤ 300 - Results for Linear Relaxation # solved %GAP Time (s) Form. instances Avg. Std. Dev. Avg.(∗) FMDrcut 16 0.070 0.088 1371.81 FMDsec 15 0.070 0.088 1828.16 16 EUC FMDrflo 16 8.448 1.120 48.57 Instances FMDrmtz (14) 16 5.060 1.179 9.62 FMDrmtz (15) 16 5.060 1.179 11.78 FMDrcut 16 0.024 0.040 3149.00 FMDsec 14 0.024 0.040 1259.23 16 NEU FMDrflo 16 7.133 2.074 49.51 16 0.175 0.169 24.88 Instances FMDrmtz (14) FMDrmtz (15) 16 0.175 1.169 32.65

the ILP formulations. Integer Formulation # solved Time (s) instances Avg.(∗) 16 1826.83 14 1310.34 0 0 0 16 3169.98 13 400.31 0 14 1025.87 14 884.33

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Table 5: EUC and NEU Instances with c ≥ 400 - Results for the ILP formulations. Linear Relaxation (g = 1, 2) Linear Relaxation (g = 3, 4) Integer Formulation solved %GAP Time (s) solved % GAP Time (s) # solved Time (s) Form. inst. Avg. Std.Dev. Avg. inst. Avg. Std.Dev. Avg. instances Avg.(∗) FMDrcut 0 9.950 0.365 2 7.023 3.315 2 FMDsec 2 2.087 1.530 0 13.595 8.260 2 24 EUC FMDrmtz (14) 12 4.148 0.321 29.10 12 4.838 1.263 808.44 0 Instances FMDrmtz (15) 12 4.148 0.321 35.22 12 4.838 1.263 859.59 0 FMDrflo 12 10.004 0.304 1437.60 12 7.888 1.240 1853.82 0 FMDrcut 0 5.883 0.984 0 9.728 3.211 0 FMDsec 0 1.253 0.380 0 3.554 2.712 0 24 NEU FMDrmtz (14) 12 0.051 0.076 188.07 12 2.930 2.732 1499.30 9 4222.01 Instances FMDrmtz (15) 12 0.051 0.076 437.57 12 2.930 2.732 1381.39 13 2541.85 FMDrflo 12 5.925 1.014 1317.23 12 9.781 3.412 3066.72 0 (*) Considering the solved instances only.

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For these EUC and NEU larger instances, we can observe that, within the time limit, FMDrcut and FMDrflo produce comparable average percentage gaps (slightly smaller for g = 3, 4) with similar standard deviation. FMDsec obtains larger gaps in the EUC instances from group g = 3, 4 (13.6% on average) and much smaller gaps in the EUC instances from groups g = 1, 2 and in the NEU instances (averages less than 3.5%). Regarding FMDrmtz , it performs even better than FMDsec in the NEU instances, specially in the groups g = 1, 2, where the average percentage gap is 0.05. In the EUC instances, it yields an average percentage gap less than 5%, which is better than FMDrcut and FMDrflo , and considerably better than FMDsec for g = 3, 4. Regarding the computational effort, the linear relaxations of FMDrcut and FMDsec always exceed the time limit or the memory limit, for c ≥ 500. On the other hand, the linear relaxation of FMDrmtz is often solved within less that 1h. Although still reasonable, the processing times for the linear relaxation of FMDrflo are always higher. Only 4 out of these 24 larger EUC instances could be solved to optimality (2 by FMDrcut and 2 by FMDsec ). On the other hand, the small gaps produced by FMDrmtz in reasonable processing times allow it to find optimal solutions for many NEU instances. Using either (14) or (15), not the same instances were solved. The first alternative solved 2 instances that the second one was not able to solve, whereas 6 is the score in favor of the second alternative. The different performances, both in terms of the solved instances as well as in terms of the processing times, are due to the moment the feasible solutions are found by CPLEX. We also remark that, if we apply the original MTZ constraints (13) instead, the times required by the linear relaxations decrease 80% approximately but the average percentage gaps more than double, which makes it prohibitive to solve the integer formulation in the time limit. 6.3. Lagrangian Algorithms

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Now we analyze the computational performance of three versions of Algortihm 1, to be denoted LA1, LA2 and LA3. Algorithm LA1 (resp. LA2, LA3) embeds Heuristic LH1 (resp. LH2, LH3). We only consider the EUC and NEU instances in this experiment, since the ALM instances were easily solved to optimality by the formulations. Again the time limit is 18,000s. As for the ILP formulations, it can be slightly exceeded because the time checking is only made after a complete iteration of the Lagrangian algorithm. We first compare the gaps returned by each algorithm independently. For this, we calculate %GAP=100(UB-LB)/LB, where UB and LB are the best upper and lower bounds provided by each algorithm separately during its own execution. We noticed that the 3 algorithms usually produce narrow gaps, even for instances that were not solved by any of the tested formulations, such as the EUC instances with c ≥ 500. Although mostly very small, the gaps produced by LA1 can go up to almost 47% (EUC) or 25% (NEU) in a few instances. These largest gaps occur when a small number of iterations is executed due to the time limit (the DSM stops before convergence). Very similar comments can be made in connection with LA2: most gaps are very small but some of them can be large (around 29% (EUC) or 15%(NEU)) in a few instances where the DSM stops before convergence. On the other hand, LA3 tends to yield individually larger gaps but within a much tighter ranges. Most of the gaps are below 5% and only one is above 10%. These data are summarized in Table 6. We can observe that every algorithm performs much better with the instances of groups g = 1, 2 than with those of groups g = 3, 4. This is particularly true for LA1 and LA2, which provided the best average percentage gaps and standard deviations for g = 1, 2. In contrast, for g = 3, 4, LA3 performed better than LA1 and LA2, specially on the EUC instances, where it got the smallest average gap overall. However, it is worth remarking that, when excluding the (not many) cases where the DSM stopped due to the time limit, the percentage gaps of the three algorithms become similar. 24

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Table 6: Gaps produced Averages for LA1 Instances g = 1, 2 g = 3, 4 Avg. StdDev. Avg. StdDev. 40 EUC 0.129 0.163 9.029 14.414 40 NEU 0.110 0.107 8.282 6.637

by each Lagrangian algorithm Averages for LA2 g = 1, 2 g = 3, 4 Avg. StdDev. Avg. StdDev. 0.114 0.082 5.984 8.781 0.230 0.410 6.958 5.010

independently. Averages for LA3 g = 1, 2 g = 3, 4 Avg. StdDev. Avg. StdDev. 1.460 0.315 2.929 2.139 2.747 0.828 6.253 2.491

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Table 7 evaluates the quality of the feasible solutions found by each Lagrangian heuristic. We now calculate %GAP = 100(U B − BLB)/BLB, where U B is the value of the best feasible solution found by the heuristic, and BLB is the best obtained lower bound (the optimal value, if known). This table presents the averages and standard deviation of %GAP as well as the average processing times (in seconds). In general, we can observe that the obtained solutions are close to the optima, specially for g = 1, 2. LA1 and LA2 are comparable and performed better than LA3 for c ≤ 300 or g = 1, 2. For c ≥ 400 and mainly when g = 3, 4, LA3 showed the best performance. It is worth remarking that the average time demanded by LA3 is around 10% of that spent by each of the other two algorithms.

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Table 7: Quality of the feasible solutions found by the Lagrangian heuristics. LA1 LA2 LA3 %GAP StdDev. Time(s) %GAP StdDev. T(s) %GAP StdDev. Time(s) c ≤ 300 0.486 0.920 95.29 0.471 0.805 87.41 0.986 0.738 8.62 40 EUC c ≥ 400 3.705 5.463 8825.01 2.555 3.657 8575.81 2.256 1.681 901.16 Instances g = 1, 2 0.082 0.084 4384.54 0.090 0.081 4163.84 1.028 0.211 900.32 g = 3, 4 4.753 5.527 6281.71 3.352 3.645 6197.06 2.468 1.878 187.97 c ≤ 300 1.743 3.436 97.87 1.056 2.137 93.07 2.800 1.701 26.62 40 NEU c ≥ 400 5.093 6.142 8236.37 4.493 4.769 8914.61 3.620 2.556 1443.56 0.093 4286.93 0.140 0.189 5106.46 1.656 0.640 1345.12 Instances g = 1, 2 0.089 g = 3, 4 7.418 5.691 5675.02 6.096 4.290 5665.53 4.928 2.119 408.44

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Lastly, we compare the lower bounds given by the ILP formulations and the Lagrangian relaxations. For each instance, we calculate the percentage differences 100(BLB − LB)/(BLB), where LB is the lower bound obtained by a linear relaxation or a Lagrangian algorithm, and BLB is the maximum obtained lower bound. Table 8 shows the averages. For EUC instances, the Lagrangian algorithms seem to provide the best trade-off between quality of the lower bound and processing time (times can be seen in the previous tables). In particular, we can highlight the performance of LA3. Although also good in the NEU instances, LA3 is outperformed by FMDmtz . We can observe that, if the focus of the Lagrangian algorithms were the lower bound only, we could have introduced lighter feasibility/improvement procedures to allow a greater number of iterations and potentially increase the final lower bound. Table 8: Quality of lower bounds - Averages of 100(BLB − LB)/BLB. FMDrcut FMDsec FMDrmtz LA1 LA2 LA3 c ≤ 300 0.00 0.01 4.99 0.25 0.24 0.38 40 EUC c ≥ 400 6.91 6.26 2.91 2.42 1.60 0.41 Instances g = 1, 2 5.91 1.19 4.28 0.03 0.00 0.40 g = 3, 4 2.38 6.33 3.21 3.08 2.11 0.40 c ≤ 300 0.00 0.02 0.15 0.10 0.07 0.68 40 NEU c ≥ 400 6.32 0.92 0.01 0.55 0.64 1.44 Instances g = 1, 2 3.51 0.74 0.04 0.02 0.09 1.06 g = 3, 4 4.08 0.37 0.09 0.72 0.74 1.21

Some final remarks in connection with the ingredients used in the Lagrangian algorithms 25

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are worthwhile. FP1 and the Kruskal-like procedure used in LH1/LH2 and LH3, respectively, provide good feasible solutions that are only slightly increased by MINCOST-DEGREE-COMPLETION or the local search. The Kruskal-like procedure seems to be a good alternative for FP1 if the time is really a limit for the Lagrangian Algorithm. Although FP1 usually yields better solutions in the same number of iterations, the lighter alternative procedure allows more iterations. 7. Conclusion

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We introduced a variant of the Min-Degree Constrained Minimum Spanning Tree Problem where the central and terminal nodes are fixed a priori. It models, for instance, applications where one wants to design a tree network linking terminals through hubs which are required to connect at least a certain number of devices due to technical or economical reasons. We studied the computational complexity of this problem in terms of the input graph and the minimum degree requirements. We proved that the problem is NP-Hard even for complete graphs, but the feasibility version is polynomial in this case. However, this later problem remains hard even when there is an edge between each central and each terminal in the input graph. These complexity results are shown even in the d-uniform case, for d ≥ 2. Studying deeper the feasibility problem, we derived necessary and sufficient conditions for the existence of a feasible solution and for finding it in polynomial time. These conditions are based on the vertex partition between centrals and terminals as well as on the graph structure. We presented and tested five integer linear programming formulations, FMDsec , FMDint , FMDrcut , FMDrmtz , FMDrflo , which are ready adaptations of models for the minimum spanning tree problem (MST). It is worth observing that the MST constraints are applied only to G(C) instead of G, thus reducing the size of the formulation. We compared the performances of these formulations (considering a processing time limit of 5h for each tested instance). FMDsec and FMDrcut provided very good lower bounds (average gap less than 0.01%) within the time limit for instances up to 300 central nodes. In these cases, we could get the optimal solution. For larger instances, the percentage gap obtained in the time limit significantly grows due to a premature stop of the cutting plane process. FMDrmtz provided average gap around 5%, for Euclidean instances with c ≥ 400, or 1.5%, for Euclidean instances with c ≤ 300 and non-Euclidean instances. It could solve to optimality most instances in this second group, including non-Euclidean instances with up to 800 centrals. FMDint and FMDrflo (as well as the multicommodity-based version) did not present results as good as the other three formulations. Once we got hard instances to solve to optimality, we proposed three Lagrangian algorithms that provide both lower and upper bounds. LA3 seems to give the best trade-off between the quality of the bounds and the processing time. All of them could provide tight gaps even for instances not solved by any of the formulations. Besides its own relevance, FMD-MST can be viewed as a restriction of MD-MST that could be used as a subproblem in a resolution process (for instance, in a Benders decomposition or to provide feasible solutions). The very good computational results obtained with the ALM instances, where the existing approaches for MD-MST already encounters some difficulties, is an encouragement to proceed in this direction in a future work. Acknowledgements

This research was supported by CNPq (Proc. 308421/2013-2, 443747/2014-8) and FUNCAP (Proc. INC-0083-00047.01.00/13). We would like to thank the referees for the useful suggestions that certainly helped us to improve the paper. 26

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References

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Table A.9: Optimal solutions obtained by the ILP Formulations. T/M=time limit/memory exceeded. Instance Time (s) NEU OPT FMDrcut FMDsec FMDrmtz (15) FMDrmtz (14) 60-1 3032 2.71 1.46 0.46 0.55 60-2 3872 1.89 0.16 1.56 0.97 60-3 2972 4.99 1.27 1.47 1.02 60-4 4031 1.05 0.56 5.01 7.99 100-1 3079 16.91 13.04 4.43 1.41 100-2 3858 21.52 20.10 5.70 4.09 Instance Time (s) 100-3 3084 16.45 5.89 12.92 9.22 EUC OPT FMDrcut FMDsec 100-4 3655 13.94 15.06 312.50 384.46 60-1 6943 3.10 1.45 200-1 3326 993.73 388.16 43.79 16.51 60-2 6772 3.29 0.88 200-2 4280 906.18 314.58 1444.25 2058.99 60-3 6709 2.09 1.25 200-3 3388 489.28 222.29 744.91 1326.11 60-4 7040 2.12 1.33 200-4 4223 709.30 854.86 M M 100-1 8217 18.01 13.72 300-1 3462 17709.72 T 119.36 2580.86 100-2 8069 15.65 7.23 300-2 4382 14111.02 T 402.90 6599.00 100-3 8055 14.10 10.94 300-3 3326 7708.38 3366.65 9281.42 1370.94 100-4 8139 11.67 22.49 300-4 4001 8012.62 M T T 200-1 11509 715.47 117.05 400-1 3547 T T 742.20 15106.18 200-2 11305 936.63 248.80 400-2 4220 T T 733.38 7220.58 200-3 11454 806.21 3856.10 400-3 T T T T 200-4 11391 2795.47 9031.79 400-4 T T T T 300-1 13463 6928.37 2870.33 500-1 3510 T T 419.13 148.13 300-2 13004 7213.40 2161.36 500-2 4360 T T 376.33 556.06 300-3 12797 5555.20 M 500-3 T T T T 300-4 12545 4208.55 T 500-4 4322 T T 1644.48 1516.53 400-1 15518 T 9384.88 600-1 3663 T T 879.19 1212.99 400-2 15116 T 12121.26 600-2 4544 T T 2743.78 M 400-3 15397 13982.72 T 600-3 3472 T T 3354.97 4935.26 400-4 15255 10499.41 T 600-4 T T M M 700-1 3809 T T M 1207.72 700-2 4758 T M 1076.31 M 700-3 3677 T M M 6094.65 700-4 4564 T T 8174.96 M 800-1 3943 T T 1634.48 M 800-2 4936 T M 632.51 M 800-3 3709 M T 10632.29 M 800-4 T T M M

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Table A.10: Upper bounds - Percentage differences (U B − BU B)/BU B. EUC NEU Instance BUB LA1 LA2 LA3 BUB LA1 LA2 LA3 60-1 6943∗ 0.00 0.00 0.88 3032∗ 0.00 0.00 1.35 60-2 6772∗ 0.00 0.00 1.06 3872∗ 0.00 0.00 1.34 60-3 6709∗ 0.00 0.00 0.07 2972∗ 0.00 0.03 3.67 60-4 7040∗ 0.00 0.00 0.00 4031∗ 0.00 0.00 1.04 100-1 8217∗ 0.00 0.05 0.88 3079∗ 0.00 0.00 1.40 100-2 8069∗ 0.00 0.00 0.98 3858∗ 0.00 0.03 2.51 100-3 8055∗ 0.00 0.00 0.04 3084∗ 0.36 0.36 3.66 100-4 8139∗ 0.00 0.11 0.17 3655∗ 1.64 0.33 3.31 200-1 11509∗ 0.00 0.00 0.70 3326∗ 0.00 0.03 1.14 200-2 11305∗ 0.12 0.11 1.58 4280∗ 0.05 0.09 1.59 200-3 11454∗ 0.72 1.63 1.40 3388∗ 3.45 2.95 3.72 200-4 11391∗ 2.17 1.55 1.80 4223∗ 2.65 3.10 4.29 300-1 13462∗ 0.04 0.07 1.13 3462∗ 0.00 0.00 1.16 300-2 13004∗ 0.00 0.00 1.01 4382∗ 0.00 0.02 2.92 300-3 12797∗ 2.58 2.22 1.26 3326∗ 12.00 8.00 7.22 300-4 12545∗ 2.14 1.80 2.83 4001∗ 7.75 1.95 4.47 400-1 15518∗ 0.03 0.03 0.85 3547∗ 0.06 0.00 1.78 400-2 15116∗ 0.02 0.02 0.82 4220∗ 0.05 0.12 1.54 400-3 15397∗ 2.12 2.12 2.16 3704 5.86 5.59 0.00 400-4 15255∗ 2.62 2.14 1.76 4322 0.00 1.06 0.88 500-1 17320 0.00 0.02 0.69 3510∗ 0.14 0.17 1.20 500-2 17161 0.00 0.08 0.84 4360∗ 0.09 0.09 1.63 500-3 17288 0.68 0.00 0.66 3669 2.94 3.54 0.00 500-4 17939 0.99 0.00 0.43 4322∗ 4.84 5.48 2.78 600-1 18983 0.00 0.06 0.95 3663∗ 0.14 0.27 2.21 600-2 18653 0.03 0.00 0.96 4544∗ 0.26 0.35 2.90 600-3 19722 1.27 0.32 0.00 3472∗ 13.05 7.92 9.50 600-4 19651 1.73 0.00 0.20 4620 1.65 3.12 0.00 700-1 20397 0.00 0.10 1.09 3812 0.00 0.08 0.79 700-2 19935 0.00 0.06 1.08 4758∗ 0.11 0.06 1.85 700-3 21216 1.48 0.00 0.02 3677∗ 13.27 10.69 9.57 700-4 21084 4.59 1.60 0.00 4564∗ 7.27 5.78 3.83 800-1 21617 0.00 0.01 0.71 3943∗ 0.25 0.08 2.03 800-2 21326 0.03 0.00 0.98 4936∗ 0.20 0.24 0.63 800-3 23964 3.36 1.28 0.00 3709∗ 15.93 10.76 5.63 800-4 26246 10.44 4.58 0.00 4884 0.37 4.12 0.00 900-1 22825 0.03 0.00 1.02 4154 0.00 0.19 2.02 900-2 22207 0.15 0.00 1.16 5049 0.00 0.53 0.69 900-3 27329 9.81 4.97 0.00 4181 12.41 6.70 0.00 900-4 26701 9.63 5.19 0.00 5078 6.26 3.98 0.00 (∗ ) Optimal value.

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LA1 0.03 0.01 0.05 0.02 0.03 0.01 0.03 0.21 0.02 0.00 0.31 0.25 0.01 0.00 0.63 0.43 0.01 0.01 0.72 0.32 0.02 0.02 0.61 0.40 0.02 0.02 0.88 0.38 0.02 0.01 0.82 0.54 0.02 0.02 0.95 0.48 0.02 0.12 4.31 2.68

LA2 0.03 0.02 0.07 0.01 0.01 0.00 0.04 0.15 0.01 0.01 0.20 0.15 0.00 0.00 0.49 0.30 0.01 0.01 0.61 0.29 0.03 0.03 0.45 0.39 0.03 0.03 0.59 0.37 0.03 0.02 0.74 0.45 0.02 0.03 0.88 2.72 0.52 0.96 0.84 5.68

LA3 0.29 0.25 0.25 0.44 0.39 0.30 0.52 0.76 0.51 0.90 1.15 1.10 0.95 0.94 1.38 1.11 1.09 1.13 1.39 1.17 1.45 1.24 1.59 1.23 1.53 1.27 1.77 1.39 1.44 1.19 1.67 1.42 1.66 1.34 1.67 1.57 1.74 1.58 1.68 1.55

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Percentage differences (BLB − LB)/BLB. NEU LA2 LA3 BLB cut sec mtz flo 0.01 0.08 3032∗ 0.00 0.00 0.00 8.65 0.01 0.12 3872∗ 0.00 0.00 0.05 5.15 0.01 0.00 2972∗ 0.05 0.05 0.08 8.44 0.01 0.01 4031∗ 0.00 0.00 0.32 4.70 0.09 0.30 3079∗ 0.00 0.00 0.00 8.79 0.04 0.25 3858∗ 0.00 0.00 0.16 4.40 0.02 0.01 3084∗ 0.02 0.02 0.09 11.67 0.30 0.10 3655∗ 0.14 0.14 0.56 4.70 0.01 0.34 3326∗ 0.00 0.00 0.00 7.43 0.16 0.55 4280∗ 0.00 0.00 0.33 6.21 1.27 1.23 3388∗ 0.04 0.04 0.20 8.37 1.78 2.10 4223∗ 0.03 0.03 0.46 5.57 0.04 0.40 3462∗ 0.00 0.32 0.09 7.67 0.01 0.33 4382∗ 0.00 0.01 0.07 5.73 0.45 0.49 3326∗ 0.09 0.09 0.12 9.74 0.75 0.90 4001∗ 0.02 0.02 0.26 6.90 0.03 0.51 3547∗ 6.90 0.47 0.06 7.14 0.06 0.50 4220∗ 5.75 0.92 0.09 5.96 0.66 0.72 3487.63 8.42 0.24 0.00 8.70 0.65 0.43 4188.33 6.09 0.08 0.00 6.31 0.00 0.39 3510∗ 7.47 1.15 0.00 7.52 0.00 0.43 4360∗ 5.22 1.11 0.00 5.26 0.00 0.07 3443.52 8.05 0.36 0.00 8.11 0.00 0.10 4322∗ 5.44 0.64 0.04 5.47 0.00 0.49 3663∗ 6.63 1.24 0.00 6.65 0.00 0.52 4544∗ 5.04 0.90 0.02 5.06 0.01 0.00 3472∗ 8.49 0.49 0.04 8.50 0.00 0.06 4424.20 6.91 0.85 0.00 6.91 0.00 0.47 3809.00 6.67 1.31 0.00 6.67 0.00 0.54 4758.00 5.32 1.56 0.00 5.31 0.00 0.03 3677∗ 8.15 0.83 0.04 8.14 0.44 0.00 4564∗ 5.39 0.94 0.03 5.36 0.00 0.44 3943∗ 6.74 1.55 0.00 6.71 0.00 0.52 4936∗ 4.48 1.24 0.00 4.48 1.45 0.00 3709∗ 7.11 0.59 0.05 7.10 11.07 0.99 4704.75 5.33 0.93 0.00 5.32 0.00 0.73 4150.00 5.55 1.69 0.00 5.54 0.00 0.58 5036.00 4.39 1.47 0.00 4.37 11.41 0.00 3936.24 7.32 0.84 0.00 7.28 12.89 1.57 4856.71 5.08 0.90 0.00 -

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BLB 6943∗ 6772∗ 6709∗ 7040∗ 8217∗ 8069∗ 8055∗ 8139∗ 11509∗ 11305∗ 11454∗ 11391∗ 13462∗ 13004∗ 12797∗ 12545∗ 15518∗ 15116∗ 15397∗ 15255∗ 17291.08 17153.01 16951.51 17626.66 18964.28 18646.02 19295.69 19192.42 20375.68 19894.24 20697.79 20168.66 21592.29 21303.68 23130.54 24752.63 22782.61 22179.60 25916.65 25175.75

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Inst. 60-1 60-2 60-3 60-4 100-1 100-2 100-3 100-4 200-1 200-2 200-3 200-4 300-1 300-2 300-3 300-4 400-1 400-2 400-3 400-4 500-1 500-2 500-3 500-4 600-1 600-2 600-3 600-4 700-1 700-2 700-3 700-4 800-1 800-2 800-3 800-4 900-1 900-2 900-3 900-4

Table A.11: Lower bounds EUC cut sec mtz flo LA1 0.00 0.00 6.52 8.40 0.01 0.00 0.00 5.54 7.95 0.00 0.00 0.00 7.20 9.30 0.01 0.00 0.00 7.23 7.23 0.01 0.09 0.09 4.02 9.09 0.09 0.04 0.04 3.20 8.29 0.04 0.00 0.00 5.77 7.49 0.04 0.00 0.00 5.18 7.02 0.21 0.00 0.00 4.21 10.16 0.00 0.15 0.15 4.73 10.33 0.16 0.18 0.18 5.51 8.01 1.00 0.25 0.25 4.85 7.09 2.23 0.04 0.04 4.60 9.61 0.04 0.01 0.01 4.57 9.83 0.01 0.14 0.14 3.91 7.90 0.48 0.21 0.29 3.93 7.48 0.73 9.26 0.03 3.99 9.77 0.03 9.55 0.06 4.30 10.02 0.06 0.01 2.78 2.54 5.54 0.65 0.10 3.41 2.32 5.73 0.50 10.35 0.02 4.81 10.43 0.00 10.17 0.46 4.29 10.24 0.00 6.60 4.08 3.04 6.90 0.09 5.94 6.69 2.93 6.04 0.00 9.99 1.32 4.05 10.02 0.00 10.07 2.64 4.24 10.11 0.00 7.28 8.75 3.59 7.20 0.17 5.70 8.79 2.07 5.71 0.17 9.36 2.69 3.83 9.29 0.00 10.04 2.93 4.05 9.99 0.00 5.83 10.05 2.66 5.79 0.02 5.02 9.74 1.41 4.90 0.70 9.71 3.36 3.70 9.63 0.00 10.05 3.09 3.72 9.93 0.00 3.71 13.61 0.90 3.53 2.10 2.41 19.66 0.00 2.06 14.53 9.79 3.59 3.78 9.69 0.02 9.87 3.66 3.81 9.75 0.46 2.69 19.77 0.05 2.46 18.11 2.44 19.27 0.00 2.26 20.75

(∗ ) Optimal value.

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