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The multi-weighted Steiner tree problem: A reformulation by intersection
Computers & Operations Research 35 (2008) 3599 – 3611 www.elsevier.com/locate/cor
The multi-weighted Steiner tree problem: A reformulation by intersection Luis Gouveia∗ , João Telhada DEIO-CIO, Faculdade de Ciências da Universidade de Lisboa, Bloco C6-Campo Grande, Cidade Universitária, 1649-016 Lisboa, Portugal Available online 13 March 2007
Abstract We propose a new formulation for the multi-weighted Steiner tree (MWST) problem. This formulation is based on the fact that a previously proposed formulation for the problem is non-symmetric in the sense that the corresponding linear programming relaxation bounds depend on the node selected as a root of the tree. The new formulation (the reformulation by intersection) is obtained by intersecting the feasible sets of the models corresponding to each possible root selection for the underlying directed problem. Theoretical results will show that the linear programming relaxation of the new formulation dominates the linear programming relaxation of each of the rooted formulations and is comparable with the linear programming bounds of the best formulation known for the problem. A Lagrangean relaxation scheme derived from the new formulation is also proposed and tested, with quite favourable results, on instances with up to 500 nodes and 5000 edges. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Network design; Spanning trees and Steiner trees; Linear programming relaxation; Reformulation techniques; Lagrangean relaxation
1. Introduction Let G = (V , E) be an undirected graph with its node set V partitioned into two subsets, P and S. The set P is the set of primary nodes and S is the set of secondary nodes. To each edge e ∈ E, we associate two positive costs, ce1 and ce2 , representing two mutually exclusive choices for installing one of the two available links representing different technologies such as optical fibre and copper cables. The two types of links will be denoted by primary links and secondary links. Primary links are more expensive than secondary links, that is, for each e ∈ E we have ce1 > ce2 . The objective is to find a minimum cost connected network such that each pair of primary nodes is connected by a path with primary links. Additionally, any two secondary nodes, or a secondary node and a primary node, can be connected by a path using either primary or secondary links. Fig. 1 depicts an example of such a two-level spanning tree with three primary nodes and four secondary nodes. Primary links are represented with thicker lines. This problem is known as the multi-weighted Steiner tree (MWST) problem, or the two-level network design problem, and has been introduced by Current et al. [1] where P was restricted to a two-node set. This problem was later studied by Duin and Volgenant [2], where some reduction tests were proposed. More recently, the relationship between alternative models (e.g., directed and undirected models) was examined in Balakrishnan et al. [3], where it was pointed out that when the edge costs are positive, any optimal solution for this problem can be characterized in the following way: (i) ∗ Corresponding author. Tel.: +351 21 7500409; fax: +351 21 7500081.
E-mail addresses: [email protected] (L. Gouveia), [email protected] (J. Telhada). 0305-0548/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2007.03.003
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Fig. 1. Example of a two-level network.
the set of primary and secondary links is a tree spanning all nodes and (ii) the subset of primary links is a Steiner tree on the subset P of primary nodes (see Fig. 1). Thus, a formulation for the MWST problem can be obtained by combining a formulation for the minimum cost Steiner tree problem together with a formulation for the minimum cost spanning tree problem (see [4] for formulations on both problems). Previously, Gouveia and Telhada [5] proposed a formulation for the MWST problem which contains constraints from a Steiner tree formulation by Khoury et al. [6] to guarantee primary connectivity (that is, to guarantee that all primary nodes are connected in a primary level network). This formulation can be seen as an arborescence formulation augmented with such primary connectivity constraints. Gouveia and Telhada have shown that the lower bounds given by the linear programming relaxation of the new formulation are not better than the lower bounds given by the strongest linear programming relaxation proposed by Balakrishnan et al. [7] (see [5] for some results). However, the computational results given by Gouveia and Telhada show that in many instances the difference between the two values is quite small. Furthermore, for several instances, CPU times needed to solve the linear programming relaxation of the new formulation were smaller than the CPU times needed to obtain the linear programming relaxation optimum value of the Balakrishnan et al. [7] model. The fact that the new formulation is not symmetric (that is, its linear programming relaxation value depends on the choice of the node to be the root of the tree) gives the theoretical motivation for this work. More precisely, it motivates a new formulation whose linear programming relaxation bound is at least as good as the linear programming relaxation bound obtained by using the previous formulation for all possible root selections. The framework for deriving the new formulation is given in Section 3. In Section 2 we review some important notation along with relevant results to the following sections. A Lagrangean relaxation scheme derived from the new formulation is presented in Section 4. Computational results are reported in Section 5. 2. Notation and results review As mentioned in the previous section, G = (V , E) is used to denote an undirected graph. The set of edges with only one end node in a subset of nodes U is denoted by (U ). In particular, when U is a singleton, ({k}) represents the set of edges incident in node k. If (P) is a valid formulation for the problem, (PL ) denotes the corresponding linear programming relaxation obtained by replacing the binary constraints by constraints stating that each variable varies between 0 and 1. The optimum value of (P) (respectively, (PL )) is denoted by v ∗ (P) (respectively, by v ∗ (PL )). The set of feasible solutions for formulation (P) is denoted by F (P), (P) being either a linear programming or mixed integer programming problem. If (P) is defined on an extended set of variables (x, y), then the projection of its feasible solutions set onto the set of variables x is denoted by PROJ{x} {F (F)}. It is well known that directing a network design problem yields better models, in what concerns the lower bounds obtained by using the corresponding linear programming relaxations. See, for example, Magnanti and Wolsey [4] on directed models for the minimum spanning tree problem. In this case, we consider an arc set A which contains two arcs
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(i, j ) and (j, i) for each edge {i, j } in E with the same cost as the original edge. Directing the problem also requires choosing a root node. In the context of the MWST problem, the root must always be a primary node and will usually be referred to as r. Any arc incident into such node need not be considered in the directed model. Following the observations given in the Introduction in terms of the original undirected graph, modelling a directed MWST involves characterizing the set of arcs where primary equipment is to be installed and the set of arcs where secondary equipment is to be installed (see [5,7] for more details). Therefore, we shall use the two sets of variables, xij1 and xij2 , which indicate whether arc (i, j ) is being used in the directed version of the problem, with primary or secondary equipment, respectively. k (U ) = (i,j )∈A:i,j ∈U x k , k = 1, 2, and X12 (U ) = To simplify notation used in the remainder of the text, X ij 1 + x 2 ). For each cutset induced by a node subset U , X k (U C , U ) and X 12 (U C , U ) follow the (x (i,j )∈A:i,j ∈U ij
ij
same pattern, where U C denotes the complementary set of U in V . The formulation by Gouveia and Telhada [5] (and denoted by (AArbr ), where r is the root node associated with the corresponding directed problem) is as follows: 1 1 2 2 (AArbr ) Min (cij xij + cij xij ), (i,j )∈A
1 (V − {j }, {j }) = 1, X
∀j ∈ P − {r},
12 (V − {j }, {j }) = 1, X 12 (U )|U | − 1, X
∀j ∈ S,
(2)
∀U ⊆ V − {r} : |U |2,
1 (V − {i, j }, {i})xij1 , X xij1 , xij2 ∈ {0, 1},
(1)
∀(i, j ) ∈ A : i ∈ S,
∀(i, j ) ∈ A.
(3) (4) (5)
This formulation has o(2|A|) variables and o(|V | + 2|V | + |A|) constraints. Connectivity is guaranteed by the indegree constraints (1) and (2) together with the subtour elimination constraints (3). Constraints (4) guarantee primary connectivity. Following Magnanti and Wolsey [4], a compact formulation providing the same linear programming bound can be obtained by replacing constraints (3) by a network flow subproblem. The formulation thus obtained is denoted by (AArbFlowr ): 1 1 2 2 (AArbFlowr ) Min (cij xij + cij xij ), (i,j )∈A
(1), (2), (4), (5), yjki −
j ∈V −{k}:(j,i)∈A
i∈V −{k}:(i,k)∈A
j ∈V :(i,j )∈A
k yik = 1,
yijk xij1 + xij2 , yijk 0,
yijk = 0,
∀k ∈ V − {r}, i ∈ V − {r, k},
∀k ∈ V − {r},
∀(i, j ) ∈ A, k ∈ V − {r} : i = k,
∀(i, j ) ∈ A, k ∈ V − {r} : i = k.
(6)
(7) (8) (9)
The compact formulation is attractive in the sense that, for medium sized problems, one can use available MIP packages to obtain the optimum value. In fact, this formulation is used to provide the results with up to 150 nodes given in Section 5 as well as the solutions for the example given in Fig. 2. For larger problems, one can follow Gouveia and Telhada and derive the following Lagrangean relaxation which is defined as follows: (i) relax the primary constraints and (ii) solve the Lagrangean subproblem which corresponds to a minimum cost arborescence problem (see [5] for more details).
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Fig. 2. Example of an instance that illustrates the non-symmetry of (AArbr ).
As noted before, a formulation is said to be symmetric if its linear programming relaxation optimum value does not change when the ordering of nodes is changed (see [8]). Changing the root node gives an example of reordering the nodes of the graph. It is easy to verify that (AArbr ) is not symmetric (see Fig. 2). In the upper part of Fig. 2 an instance of the MWST problem is depicted. Associated to each edge of that graph is a pair of costs (primary and secondary, respectively). In the lower left part of the same figure, the linear programming relaxation optimal solution of (AArbr ), for the case when r = 1, is presented. The linear programming relaxation optimal solution for the case when r = 6 is presented in the lower right part of Fig. 2. In both solutions, the values next to each arc indicate the values of the corresponding primary and secondary variables, xij1 and xij2 . Primary values are represented by double stroke lines whereas secondary ones are represented by simple lines. Note that the optimal values, for each of the two linear programming relaxations, differ. For the case when r = 1, that value is equal to 11.5, while for the case when r = 6 that value is equal to 15.8. This example shows that different root nodes may lead to different linear programming relaxation bounds. Unfortunately, there seems to be no apparent way of selecting a priori the root that yields the best bound. This observation gives the motivation for the reformulation technique presented in the next section. Before presenting the new formulation, we note for completeness that the directed multicommodity flow formulation presented by Balakrishnan et al. [7] (denoted by BMM in the remainder of the text) is symmetric and it differs from the compact version of the augmented formulation given before, only in the primary connectivity constraints (see below): (BMM)
(AArbFlowr ) yijk xij1 + xij2 , ∀(i, j ) ∈ A, k ∈ V − {r} 1 (V − {i, j }, {i})x 1 , ∀(i, j ) ∈ A : i ∈ X ij
S
yijk xij1 + xij2 , ∀(i, j ) ∈ A, k ∈ S yijk xij1 , ∀(i, j ) ∈ A, k ∈ P − {r}
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Gouveia and Telhada [5] have shown that v ∗ (BMML ) v ∗ (AArbkL ), for all k ∈ P . As noted before, computational experience reported in that same paper shows, however, that the difference between the two linear programming bounds appears to be quite small. Furthermore, for some instances, the CPU times involved in obtaining the linear programming bounds are better for the case of the (AArbFlowr ) formulation when compared with (BMM). 3. Reformulation by intersection The basic idea for the new formulation is to consider, at the same time, the previous formulation (AArbr ) for all possible values of r, and guarantee that the feasible solutions of the models associated to every root intersect each other, in the sense that a feasible solution for a model associated to a given root node r cannot use an arc which is not used (in the same direction or in the reversed direction) by the solution of the model associated to any other root. Let Ak (E) denote the set of arcs associated to root k. In order to guarantee that each directed solution is superimposed with the remaining ones, our model will use 1 and x 2 , to define the inclusion of edge {i, j }, respectively, with primary and additional undirected variables, x{ij } {ij } secondary equipment, in the solution. These variables are introduced in order to guarantee that each directed rooted solution matches with the undirected solution (or more precisely, a feasible undirected solution is a projection of a directed solution, for each possible root). Let { xij1k }(i,j )∈Ak (E),k∈P and { xij2k }(i,j )∈Ak (E),k∈P be the variables defining primary or secondary equipment, respectively, in the model whose root node is k. In order to guarantee that each directed solution matches with the same undirected solution, the following constraints must be included in the new model: 1 x{i,j ij1k + xj1ki , }=x 1 1k kj , x{k,j }=x
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P ,
2 ij2k + xj2ki , x{i,j }=x 2k 2 , kj x{k,j }=x
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P .
(10) (11) (12) (13)
The new formulation (denoted by SuperAArb) is completed by considering the constraints from all models (AArbk ) associated to the different root nodes k together with the previously given “intersection” constraints. Note that the objective function involves, now, the undirected variables, despite the fact that it could be written in terms of any of the directed variables set: 1 1 2 2 (SuperAArb) Min (c{i,j } x{i,j } + c{i,j } x{i,j } ), {i,j }∈E
1k (V − {j }, {j }) = 1, X
∀j ∈ P − {k}, k ∈ P ,
12k (V − {j }, {j }) = 1, X 12k (U )|U | − 1, X
∀j ∈ V − {k}, k ∈ P ,
(15)
∀U ⊆ V − {k}: |U |2, k ∈ P ,
(16)
1k (V − {i, j }, {i}) x1k , X ij 1 ij1k + xj1ki , x{i,j }=x 1k 1 kj , x{k,j }=x
2 2k kj , x{k,j }=x
∀(i, j ) ∈ Ak (E): i ∈ S, k ∈ P ,
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P ,
2 ij2k + xj2ki , x{i,j }=x
(14)
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P ,
(17) (10) (11) (12) (13)
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xij1k , xij2k 0,
∀(i, j ) ∈ Ak (E), k ∈ P ,
xij1 , xij2 ∈ {0, 1},
(18)
∀{i, j } ∈ E.
(19)
This new formulation has O(|E| + |P | × |A|) variables and O(|P | × (|P | + |V | + 2|V | + |A| + 2|E|)) constraints. The intersection constraints (10)–(13) guarantee that the projection of the set of feasible solutions of the (SuperAArb) linear programming relaxation onto the space defined by variables referring to a specific root satisfies all constraints included in the respective (AArbr ) model, that is: Proposition 1. PROJ{x 1
2 {i,j } ,x{i,j } }
{F (SuperAArbL )} ⊆ PROJ{x 1
2 {i,j } ,x{i,j } }
{F (AArbkL )}, for all k ∈ P .
As a consequence, the linear programming relaxation bound obtained by (SuperAArb) is at least as good as the bound obtained by any of the (AArbr ) linear programming relaxation. Proposition 2. v ∗ (SuperAArbL ) v ∗ (AArbkL ), for all k ∈ P . It remains to see what the relationship is between the linear programming relaxations of the two models, (SuperAArb) and (BMM) (which is the model with the best known linear programming bound, and already mentioned in the previous section). In all the results taken (see Section 5), the linear programming bounds of the two models are equal. However, as was shown in Telhada [9], we have that v ∗ (BMML ) v ∗ (SuperAArbL ) and, furthermore, there exist some instances for which a strict inequality holds. We skip the proof of this result as it is rather tedious. We note that it follows a similar proof given by Goemans and Myung [10]. In the same way that a network flow compact version of the formulation (AArbr ) was derived in [5], we can also obtain a compact version of the formulation (SuperAArb) by replacing the subtour elimination constraints by adequate flow constraints. However, as we shall show next, the intersection constraints allow us to use commodities defined only from a given primary source to every secondary node. To see this, first note that for each k, the directed subtour elimination constraints (16) are equivalent, under the presence of intersection constraints (10)–(13), to a unique set of undirected subtour elimination constraints. X12 (U )|U | − 1,
∀U ⊆ V : |U |2.
(20)
This result allows us to replace the directed subtour elimination constraints (16) by the more compact set of undirected subtour elimination constraints (20) in the formulation (SuperAArb). This modified formulation will be used as the starting point for deriving a Lagrangean relaxation scheme (see Section 4). For the purpose of providing the compact formulation, we shall additionally show that some of constraints (20) are redundant and thus, can be eliminated: 1 , x 2 , x 1k , x 2k } that satisfies (15), (10)–(13) and (18) also satisfies constraints (20) Proposition 3. A solution {x{i,j ij } {i,j } ij for subsets U that contain at least one primary node.
Proof. Consider a subset of nodes U that contains at least one primary node. Let k ∗ be a primary node in U . Using the intersection constraints (10)–(13) for node k ∗ , 12k ∗ (U − {i}, {i}). 12k ∗ (U ) = X X12 (U ) = X i∈U −{k ∗ }
Non-negativity constraints (18) guarantee that 12k ∗ (U − {i}, {i}) 12k ∗ (V − {i}, {i}). X X i∈U −{k ∗ }
i∈U −{k ∗ }
Using the indegree constraints (15) gives 12k ∗ (V − {i}, {i}) = X X 12 (U ) i∈U −{k ∗ }
i∈U −{k ∗ }
1 = |U | − 1.
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Fig. 3. Optimal solution for (SuperAArbL ) without the subtour elimination constraints.
The previous proposition shows that we need to consider subtour elimination constraints (20) in formulation (SuperAArb) only for subsets U that are contained in S. To highlight the need of using the remaining subtour elimination constraints to obtain a valid model for the problem, Fig. 3 shows the optimal solution produced by (SuperAArb) without the whole set of subtour elimination constraints for the same instance shown in Fig. 2. This solution is unfeasible, as it contains a subtour involving only nodes in S. Before obtaining the equivalent compact formulation, we note that by using the intersection constraints, for a given 12k ∗ (U ), and then by using the indegree constraints (15), we know primary node k ∗ , which guarantee that X 12 (U ) = X that the subtour elimination constraints (16) can be rewritten in equivalent cut form as follows: 12k ∗ (U C , U )1, X
∀U ⊆ S: |U |2 for some k ∗ ∈ P .
(21)
This observation shows that we can obtain a formulation with an equivalent linear programming bound by replacing the subtour elimination constraints (20) for sets U contained in S (or the cut constraints (21) for the same sets) in formulation (SuperAArb) by a restricted network flow system that only contains a commodity from any given primary node k ∗ to each secondary node, leading to the model (SuperAArbFlow): 1 2 2 1 (c{i,j (SuperAArbFlow) Min } x{i,j } + c{i,j } x{i,j } ), {i,j }∈E
(10).(15), (17).(19), yjki −
j ∈V −{i,k ∗ }
j ∈V −{i,k}
j ∈V −{k}
yjkk = 1, ∗
yijk = 0,
∀i ∈ V − {k ∗ , k}, k ∈ S,
∀k ∈ S, ∗
0 yijk xij1k + xij2k ,
(22) (23)
∗
∀(i, j ) ∈ Ak (E), k ∈ S.
(24)
As before, results obtained by using such a formulation will be reported in the Computational results section for medium sized instances. This compact version was used to solve the instance of Fig. 2 (which, in turn, was used to illustrate the lack of symmetry of the model (AArbk )). For this instance, the optimal solution of (SuperAArbL ) is integer and has value 17 (see Fig. 4) and this example can be used to show that the inequality in the statement of Proposition 2 can be strict. It is interesting to compare this formulation with the one we would have obtained by applying the intersection reformulation to the compact formulation (AArbFlowr ). Such a model would contain many more commodities which were shown to be redundant by using the intersection constraints and Proposition 3.
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Fig. 4. Optimal solution of (SuperAArbL ) for the instance shown in Fig. 2.
As will be shown in the section on computational results, the new formulation appears to produce very good linear programming bounds. However, the formulation becomes difficult to use for larger instances. In order to overcome the difficulty of solving its linear programming relaxations, we describe in the next section a Lagrangean relaxation based on formulation (SuperAArb). Note that, due to similar difficulties, Balakrishnan et al. [7] developed a dual ascent method based on the model (BMM) and have reported CPU times and gaps for instances up to 500 nodes and 5000 edges. The proportion of primary nodes on the set of instances tested by Balakrishnan et al. varies between 40% and 75%. 4. A Lagrangean relaxation We propose a Lagrangean relaxation scheme derived from the new formulation. As noted before, we use the model (SuperAArb), where constraints (16) were replaced by the simpler set, X 12 (U )|U | − 1,
∀U ⊆ V − {k}: |U |2, k ∈ P ,
(20)
and add the following redundant constraint, guaranteeing that |V | − 1 edges are included in the solution: X 12 (V ) = |V | − 1.
(25)
We observe that the subproblem defined on the undirected variables can be viewed as an undirected spanning tree model. Note that the set of constraints (20) could be further reduced as explained in the previous section. However, we decided to maintain the whole set for the following two reasons: (i) we do not need to check whether the problem of selecting |V | − 1 edges with minimum cost and not containing subtours involving secondary nodes is polynomial and (ii) most importantly the relaxed problem becomes stronger if constraints (20) for all subsets are maintained in the subproblem. We can now derive a Lagrangean relaxation by dualizing the primary connectivity constraints (17) and the intersection 2k constraints (10)–(13), attaching multipliers kij to the first set of constraints and attaching 1k {i,j } and {i,j } to the second set of constraints. The relaxed problem obtained is as follows: ⎛ ⎞ 1 1 2 2 (SuperAArbLR) Min (c{i,j kij ⎝xij1k − xli1k ⎠ } x{i,j } + c{i,j } x{i,j } ) + {i,j }∈E
+
k∈P {i,j }∈E
k∈P (i,j )∈Ak (E):i∈S 1 2k 2 [1k xij1k + xj1ki − x{i,j xij2k + xj2ki − x{i,j } ) + {i,j } ( } )], {i,j } (
l∈V −{i,j }
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X 12 (V ) = |V | − 1,
(25)
1k (V − {j }, {j }) = 1, X
∀j ∈ P − {k}, k ∈ P ,
12k (V − {j }, {j }) = 1, X X (U )|U | − 1,
(14)
∀j ∈ V − {k}, k ∈ P ,
(15)
∀U ⊆ V − {k}: |U |2, k ∈ P ,
12
xij1k , xij2k 0,
(20)
∀(i, j ) ∈ Ak (E), k ∈ P ,
xij1 , xij2 ∈ {0, 1},
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(18)
∀{i, j } ∈ E.
(5)
For any given value of the multipliers, the relaxed problem can be separated into |P | + 1 subproblems: (i) an undirected spanning tree problem involving the undirected variables and which can be easily solved by the algorithm given by Prim [11], for example, and (ii) |P | semi-assignment problems which can be easily solved by inspection, one for each root node k: 1 2 2 (SuperAArbLRU ) Min (c1{i,j } x{i,j } + c{i,j } x{i,j } ), {i,j }∈E
X12 (V ) = |V | − 1,
(25)
X 12 (U )|U | − 1, xij1 , xij2 ∈ {0, 1},
∀U ⊆ V − {k}: |U |2, k ∈ P ,
(20)
∀{i, j } ∈ E,
(SuperAArbLRk ),
k∈P
Min
(5)
1k 1k 1k 2k 2k 2k 2k (c1k ij xij + cj i xj i + cij xij + cj i xj i ),
{i,j }∈Ak (E)
1k (V − {j }, {j }) = 1, X 12k
X
∀j ∈ P − {k}, k ∈ P ,
(V − {j }, {j }) = 1,
xij1k , xij2k 0,
(14)
∀j ∈ V − {k}, k ∈ P ,
(15)
∀(i, j ) ∈ Ak (E), k ∈ P .
(18)
The modified costs used in these subproblems are defined as: 1 c1{i,j } = c{i,j 1k ∀{i, j } ∈ E, }− {i,j } , k∈P 2 c2{i,j } = c{i,j }−
2k {i,j } ,
∀{i, j } ∈ E,
k∈P 1k c1k ij = {i,j } ,
c1j i = 1k {i,j } , = 1k kli , {i,j } −
∀{i, j } ∈ E : i, j ∈ P ,
c1k ij
k c1j i = 1k {i,j } + j i ,
l∈V −{i,j }
k 1k c1k ij = {i,j } + ij , k 1k c1k ij = {i,j } + ij −
c1j i = 1k {i,j } − l∈V −{i,j }
kli ,
l∈V −{i,j }
klj ,
∀{i, j } ∈ E : i ∈ S, j ∈ P , ∀{i, j } ∈ E : i ∈ P , j ∈ S,
k c1j i = 1k {i,j } + j i −
l∈V −{i,j }
klj ,
∀{i, j } ∈ E : i, j ∈ S.
To find an approximation of the optimal multipliers that produce the best lower bound value we used the method 2k proposed by Camerini et al. [12]. The multipliers = {kij , 1k {i,j } , {i,j } } are updated at each step as follows: k+1 = k + tk d k . The step length as well as the improvement directions is updated according to the Camerini–Fratta–Maffioli rule.
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At each step of the multipliers approximation procedure, we compute a feasible solution for the problem. The cost of this solution gives an upper bound on the value of the optimal solution and is obtained by examining the relaxed minimum spanning tree solution and changing (when needed) to primary the costs of the edges in paths connecting primary nodes. 5. Computational results We report our computational experience on the model and methods proposed in this paper. For our experiments, we used Euclidean cost instances and randomly generated instances (see [5] for details on those instances construction procedure). The number of nodes of the instances used varies between 50, 75, 100, 125 and 150. For each instance, we considered 25%, 50% and 75% of the nodes to be primary. For each set of parameters (number of nodes and proportion of primary nodes), 5 Euclidean instances and 5 random instances were generated. The instances with 50 nodes are the instances already used in our previous work [5]. We note, however, that the results reported here regarding the (AArb) model may be better than the ones reported earlier. This is due to some improvements on both the model and the machine used. For every generated instance, we use some of the preprocessing tests proposed by Balakrishnan et al. [3]. The computational times for these tests were negligible for the instances tested. All instances reported in this table have a 25% density. For a better understanding of the reported results, instances are divided according to the proportion of primary nodes and to the generation method. The results reported in Table 1 show the impact of the preprocessing tests. The first three columns describe each class of instances. In the first column, number of nodes (primary + secondary) and number of edges are given. The number of primary nodes is also given in percentage in the second column. The third column indicates the type of edge cost generation method used, either random or Euclidean. The last two columns give similar information to the first two columns, after reduction tests have been performed. For example, the first row gives information on instances with 50 nodes, 12 primary (which corresponds to 25% of all nodes of that instance) and 38 secondary, and 306 edges generated by the Euclidean procedure. After performing the reduction tests, instances ended up with an average of 47.8 nodes (9.8 primary and 38 secondary) and with an average of 119.6 edges. The proportion of primary nodes, after performing the reduction tests, is equal to 20%. First of all, one observes that, in almost all cases, the number of edges is significantly reduced and that the higher the percentage of primary nodes is, the higher is the reduction effect. Another interesting fact is that the reduction tests are more effective for Euclidean instances rather than random instances. However, this happens especially when the number of nodes increases. In Table 2, computational results are reported for the 3 compact formulations referred to in this paper. We used version 9 of the CPLEX MIP solver for all 3 formulations and we report the CPU time spent, in seconds, until the LP relaxation bound was obtained (and the corresponding gap), and also the CPU time until the integer optimum was obtained. All tests were performed on a Pentium 4 CPU 3.20 GHz, 512 MB of RAM machine. Instances considered have now densities varying between 10%, 25% and 40%. To observe both the effect of the number of nodes of the density on the performance of the three models, classes of instances were separated according to those parameters. The first column indicates the number of nodes and the second column indicates the density, respectively, for the group of instances considered in each line. For each model ((AArb), (SuperAArb) and (BMM)), we report CPU times to reach the linear programming relaxation optimum and the integer optimum, as well as the LP gap thus obtained. The results reported in Table 2 show that the linear programming bound of the formulation (SuperAArb) is very tight (for all the cases reported, gaps are equal to zero) and that it improves considerably on the linear programming bound of the formulation (AArb). However, the improved bounds have the same quality as the bounds given by the linear programming relaxation of the (BMM) formulation. Additionally, (BMM) takes less CPU time to obtain the linear programming bounds as well as the integer optimum values. Instances with 150 nodes and density 40% could not be solved by the (AArb) model in less than 10,000 s. Table 3 provides similar information, but now we give the results in terms of the proportion of primary nodes of the instances. The conclusions are similar to the conclusions taken from Table 2. It is interesting to point out that for instances with a larger number of primary nodes, the CPU times of the two best models are nearly identical. Unfortunately, for cases with a small number of primary nodes, the CPU times of the new model are substantially
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Table 1 Size of the instances after preprocessing Original
Reduced
N (P + S)/M
%P (%)
50(12 + 38)/306
25
50(25 + 25)/306
50
50(37 + 13)/306
75
75(18 + 57)/693
25
75(37 + 38)/693
50
75(56 + 19)/693
75
100(25 + 75)/1237
25
100(50 + 50)/1237
50
100(75 + 25)/1237
75
125(31 + 94)/1937
25
125(62 + 63)/1937
50
125(93 + 32)/1937
75
150(37 + 113)/2793
25
150(75 + 75)/2793
50
150(112 + 38)/2793
75
N(P + S)/M
%P (%)
Euclidean Random Euclidean Random Euclidean Random
47.8(9.8 + 38)/119.6 47.8(9.8 + 38)/107.8 36.2(11.2 + 25)/84 37.4(12.4 + 25)/156.8 24(11 + 13)/44 22.2(9.2 + 13)/60.8
20 20 31 33 46 40
Euclidean Random Euclidean Random Euclidean Random
70.8(13.8 + 57)/210 69.4(12.4 + 57)/507.2 56.2(18.2 + 38)/157.8 58.8(20.8 + 38)/334.2 34.2(15.2 + 19)/65.6 31.4(12.4 + 19)/77.6
19 18 32 35 44 38
Euclidean Random Euclidean Random Euclidean Random
82.4(7.4 + 75)/589.2 92.4(17.4 + 75)/257 55(5 + 50)/234.6 74(24 + 50)/176 28(3 + 25)/72.4 44.2(19.2 + 25)/75.2
9 19 9 32 11 43
Euclidean Random Euclidean Random Euclidean Random
100.4(6.4 + 94)/810.4 118(24 + 94)/316.2 68.4(5.4 + 63)/337.4 92.2(29.2 + 63)/229.4 35.2(3.2 + 32)/107.8 56(24 + 32)/101
6 20 8 32 9 42
Euclidean Random Euclidean Random Euclidean Random
122.2(9.2 + 113)/1170.8 139(26 + 113)/413.2 81.8(6.8 + 75)/460 112.8(37.8 + 75)/292 40.6(2.6 + 38)/127.4 68.6(30.6 + 38)/127.2
7 19 8 33 6 44
larger than the CPU time provided by the (BMM) model. This can be explained by the fact that, for the instances with a smaller number of primary nodes, the number of primary connectivity constraints (17) of the (SuperAArb) model, taken for each arc incident on a secondary node, is considerably higher than the number of flow linking constraints of the (BMM) model. More interesting for the new model is the information given in Table 4 that shows the results in terms of the two different methods of generating the costs for the instances. In this case, the CPU times given by the new formulation (SuperAArb) are better than the CPU times given by the (BMM) formulation for the Euclidean instances but much worse for random instances. The CPU times reported for the new model indicate that the CPU times may become critical when solving larger instances. For such cases, we suggest using, as an alternative, the Lagrangean relaxation scheme proposed in Section 4. Therefore, we also tested that procedure in order to evaluate its behaviour for instances that cannot be easily solved with the linear programming relaxation. Table 5 shows some encouraging results for 5 Euclidean instances and 5 random instances with 500 nodes and 5000 edges. For each instance, the percentage of primary nodes was considered to be equal to, respectively, 25%, 50% and 75%. Average number of iterations, average duration and average gap are again reported. In many cases, integral optimality was obtained, although in some cases a large number of iterations was needed to obtain such bounds. Nevertheless, the reported CPU times appear to be quite acceptable and are lower for Euclidean instances.
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Table 2 Results given by the three models according to number of nodes N
D (%)
AArb
SuperAArb
LP Gap (%)
CPU (s)
Int.
LP
CPU (s)
Gap (%)
BMM Int.
LP
CPU (s)
CPU (s)
Gap (%)
Int. CPU (s)
CPU (s)
50
10 25 40
0.12 0.19 0.42
0.56 0.59 1.50
0.76 1.26 4.31
0.00 0.00 0.00
0.75 0.76 1.42
0.84 0.87 1.56
0.00 0.00 0.00
0.54 0.52 1.34
0.61 0.61 1.44
75
10 25 40
0.02 0.09 0.33
4.93 8.30 14.79
5.57 42.91 204.77
0.00 0.00 0.00
6.47 16.04 15.39
6.79 18.91 16.07
0.00 0.00 0.00
5.47 9.40 10.21
5.77 15.44 10.65
100
10 25 40
0.13 0.24 0.35
16.99 8.17 40.74
187.51 26.44 230.51
0.00 0.00 0.00
45.99 25.44 77.04
46.61 26.16 79.05
0.00 0.00 0.00
19.53 9.27 30.14
20.09 10.11 31.03
125
10 25 40
0.01 0.13 0.32
50.35 35.98 75.74
96.17 76.36 3907.71
0.00 0.00 0.00
76.61 93.10 274.70
79.72 94.49 366.66
0.00 0.00 0.00
27.55 14.84 68.12
28.97 16.05 71.28
150
10 25 40
0.00 0.17 –
51.99 42.04 –
61.59 187.42 –
0.00 0.00 0.06
255.36 353.96 1604.87
257.74 475.08 3926.62
0.00 0.00 0.00
61.10 56.50 994.13
62.92 59.92 1259.65
Table 3 Results given by the three models according to the proportion of primary nodes %P (%)
AArb
SuperAArb
LP
25 50 75
Int.
LP
BMM Int.
LP
Int.
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
0.38 0.09 0.04
55.99 12.58 0.33
945.67 26.94 0.42
0.00 0.00 0.00
202.87 39.26 0.35
245.57 40.45 0.44
0.00 0.00 0.00
49.26 12.50 0.32
52.73 13.12 0.40
Table 4 Results given by the three models according to cost generation method G
AArb
SuperAArb
LP
Euclidean Random
Int.
LP
BMM Int.
LP
Int.
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
0.22 0.14
15.09 33.77
61.98 622.24
0.00 0.00
12.13 157.97
13.74 187.35
0.00 0.00
14.86 29.21
16.00 31.03
Table 5 Some results obtained for instances with 500 nodes N/M
500/5000
%P (%)
25 50 75
Euclidean
Random
Iter.
CPU (s)
Gap (%)
Iter.
CPU (s)
Gap (%)
3535 1186.6 2383.4
6353.8 464 116.2
0.078 0.000 0.002
8554.2 4491 327
7596.6 2634.6 59
0.049 0.084 0.000
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6. Conclusions Previously, Gouveia and Telhada [5] proposed a formulation for the MWST problem that is not symmetric (that is, its linear programming relaxation value depends on the choice of the node to be the root of the tree). In this paper we proposed a new formulation whose linear programming relaxation bound is at least as good as the linear programming relaxation bound obtained by using the previous formulation for all possible root selections. The basic idea for obtaining the new formulation is to consider, at the same time, the previous formulation for all possible root choices, and guarantee that the feasible solutions of the models associated to every root intersect each other. Our results show that the linear programming relaxation of this so-called “reformulation by intersection” substantially improves on the linear programming relaxation of the original model. The new lower bounds are comparable with the lower bounds of the best known formulation for the problem. We also suggested a new Lagrangean relaxation based on the new model and presented results with up to 500 nodes and 5000 edges. Reported gaps were less than 1% and are equal to zero in many cases. We conclude this paper by noting that the reformulation by intersection idea can be applied to improve the linear programming relaxation of formulations of many other discrete optimization problems, provided that the original given formulation “suffers” from lack of symmetry. Acknowledgement This work was supported by contract grant sponsor: FCT under contract Grant number: Po CTI-ISFL-1-152. References [1] Current JR, Revelle CS, Cohon JL. The hierarchical network design problem. European Journal of Operational Research 1986;27:57–66. [2] Duin C, Volgenant T. The multi-weighted Steiner tree problem. Annals of Operations Research 1991;33:451–69. [3] Balakrishnan A, Magnanti T, Mirchandani P. Modeling and heuristic worst-case performance analysis of the two-level network design problem. Management Science 1994;40(7):846–67. [4] Magnanti T, Wolsey L. Optimal trees. Network models. Handbooks in operations research and management science, vol. 7. 1995. p. 503–615. [5] Gouveia L, Telhada J. An augmented arborescence formulation for the two-level network design problem. Annals of Operations Research 2001;106(1):47–61. [6] Khoury BN, Pardalos PM, Hearn DW. Equivalent formulations for the Steiner problem in graphs. In: Du D-Z, Pardalos PM, editors. Network optimization problems. 1993. p. 111–23. [7] Balakrishnan A, Magnanti T, Mirchandani P. Dual-based algorithm for multi-level network design. Management Science 1994;40(5):567–81. [8] Yannakakis M. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 1991;43(3): 441–66. [9] J. Telhada, Desenho Topológico de Redes Hierárquicas. PhD thesis, University of Lisbon; 2004. [10] Goemans MX, Myung Y-S. A catalog of Steiner tree formulations. Networks 1993;23:19–28. [11] Prim RC. Shortest connection networks and some generalizations. Bell System Technical Journal 1957;36:1389–401. [12] Camerini PM, Fratta L, Maffioli F. On improving relaxation methods by modified gradient techniques. Mathematical Programming Study 1975;3:26–34.
The multi-weighted Steiner tree problem: A reformulation by intersection Luis Gouveia∗ , João Telhada DEIO-CIO, Faculdade de Ciências da Universidade de Lisboa, Bloco C6-Campo Grande, Cidade Universitária, 1649-016 Lisboa, Portugal Available online 13 March 2007
Abstract We propose a new formulation for the multi-weighted Steiner tree (MWST) problem. This formulation is based on the fact that a previously proposed formulation for the problem is non-symmetric in the sense that the corresponding linear programming relaxation bounds depend on the node selected as a root of the tree. The new formulation (the reformulation by intersection) is obtained by intersecting the feasible sets of the models corresponding to each possible root selection for the underlying directed problem. Theoretical results will show that the linear programming relaxation of the new formulation dominates the linear programming relaxation of each of the rooted formulations and is comparable with the linear programming bounds of the best formulation known for the problem. A Lagrangean relaxation scheme derived from the new formulation is also proposed and tested, with quite favourable results, on instances with up to 500 nodes and 5000 edges. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Network design; Spanning trees and Steiner trees; Linear programming relaxation; Reformulation techniques; Lagrangean relaxation
1. Introduction Let G = (V , E) be an undirected graph with its node set V partitioned into two subsets, P and S. The set P is the set of primary nodes and S is the set of secondary nodes. To each edge e ∈ E, we associate two positive costs, ce1 and ce2 , representing two mutually exclusive choices for installing one of the two available links representing different technologies such as optical fibre and copper cables. The two types of links will be denoted by primary links and secondary links. Primary links are more expensive than secondary links, that is, for each e ∈ E we have ce1 > ce2 . The objective is to find a minimum cost connected network such that each pair of primary nodes is connected by a path with primary links. Additionally, any two secondary nodes, or a secondary node and a primary node, can be connected by a path using either primary or secondary links. Fig. 1 depicts an example of such a two-level spanning tree with three primary nodes and four secondary nodes. Primary links are represented with thicker lines. This problem is known as the multi-weighted Steiner tree (MWST) problem, or the two-level network design problem, and has been introduced by Current et al. [1] where P was restricted to a two-node set. This problem was later studied by Duin and Volgenant [2], where some reduction tests were proposed. More recently, the relationship between alternative models (e.g., directed and undirected models) was examined in Balakrishnan et al. [3], where it was pointed out that when the edge costs are positive, any optimal solution for this problem can be characterized in the following way: (i) ∗ Corresponding author. Tel.: +351 21 7500409; fax: +351 21 7500081.
E-mail addresses: [email protected] (L. Gouveia), [email protected] (J. Telhada). 0305-0548/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2007.03.003
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Fig. 1. Example of a two-level network.
the set of primary and secondary links is a tree spanning all nodes and (ii) the subset of primary links is a Steiner tree on the subset P of primary nodes (see Fig. 1). Thus, a formulation for the MWST problem can be obtained by combining a formulation for the minimum cost Steiner tree problem together with a formulation for the minimum cost spanning tree problem (see [4] for formulations on both problems). Previously, Gouveia and Telhada [5] proposed a formulation for the MWST problem which contains constraints from a Steiner tree formulation by Khoury et al. [6] to guarantee primary connectivity (that is, to guarantee that all primary nodes are connected in a primary level network). This formulation can be seen as an arborescence formulation augmented with such primary connectivity constraints. Gouveia and Telhada have shown that the lower bounds given by the linear programming relaxation of the new formulation are not better than the lower bounds given by the strongest linear programming relaxation proposed by Balakrishnan et al. [7] (see [5] for some results). However, the computational results given by Gouveia and Telhada show that in many instances the difference between the two values is quite small. Furthermore, for several instances, CPU times needed to solve the linear programming relaxation of the new formulation were smaller than the CPU times needed to obtain the linear programming relaxation optimum value of the Balakrishnan et al. [7] model. The fact that the new formulation is not symmetric (that is, its linear programming relaxation value depends on the choice of the node to be the root of the tree) gives the theoretical motivation for this work. More precisely, it motivates a new formulation whose linear programming relaxation bound is at least as good as the linear programming relaxation bound obtained by using the previous formulation for all possible root selections. The framework for deriving the new formulation is given in Section 3. In Section 2 we review some important notation along with relevant results to the following sections. A Lagrangean relaxation scheme derived from the new formulation is presented in Section 4. Computational results are reported in Section 5. 2. Notation and results review As mentioned in the previous section, G = (V , E) is used to denote an undirected graph. The set of edges with only one end node in a subset of nodes U is denoted by (U ). In particular, when U is a singleton, ({k}) represents the set of edges incident in node k. If (P) is a valid formulation for the problem, (PL ) denotes the corresponding linear programming relaxation obtained by replacing the binary constraints by constraints stating that each variable varies between 0 and 1. The optimum value of (P) (respectively, (PL )) is denoted by v ∗ (P) (respectively, by v ∗ (PL )). The set of feasible solutions for formulation (P) is denoted by F (P), (P) being either a linear programming or mixed integer programming problem. If (P) is defined on an extended set of variables (x, y), then the projection of its feasible solutions set onto the set of variables x is denoted by PROJ{x} {F (F)}. It is well known that directing a network design problem yields better models, in what concerns the lower bounds obtained by using the corresponding linear programming relaxations. See, for example, Magnanti and Wolsey [4] on directed models for the minimum spanning tree problem. In this case, we consider an arc set A which contains two arcs
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(i, j ) and (j, i) for each edge {i, j } in E with the same cost as the original edge. Directing the problem also requires choosing a root node. In the context of the MWST problem, the root must always be a primary node and will usually be referred to as r. Any arc incident into such node need not be considered in the directed model. Following the observations given in the Introduction in terms of the original undirected graph, modelling a directed MWST involves characterizing the set of arcs where primary equipment is to be installed and the set of arcs where secondary equipment is to be installed (see [5,7] for more details). Therefore, we shall use the two sets of variables, xij1 and xij2 , which indicate whether arc (i, j ) is being used in the directed version of the problem, with primary or secondary equipment, respectively. k (U ) = (i,j )∈A:i,j ∈U x k , k = 1, 2, and X12 (U ) = To simplify notation used in the remainder of the text, X ij 1 + x 2 ). For each cutset induced by a node subset U , X k (U C , U ) and X 12 (U C , U ) follow the (x (i,j )∈A:i,j ∈U ij
ij
same pattern, where U C denotes the complementary set of U in V . The formulation by Gouveia and Telhada [5] (and denoted by (AArbr ), where r is the root node associated with the corresponding directed problem) is as follows: 1 1 2 2 (AArbr ) Min (cij xij + cij xij ), (i,j )∈A
1 (V − {j }, {j }) = 1, X
∀j ∈ P − {r},
12 (V − {j }, {j }) = 1, X 12 (U )|U | − 1, X
∀j ∈ S,
(2)
∀U ⊆ V − {r} : |U |2,
1 (V − {i, j }, {i})xij1 , X xij1 , xij2 ∈ {0, 1},
(1)
∀(i, j ) ∈ A : i ∈ S,
∀(i, j ) ∈ A.
(3) (4) (5)
This formulation has o(2|A|) variables and o(|V | + 2|V | + |A|) constraints. Connectivity is guaranteed by the indegree constraints (1) and (2) together with the subtour elimination constraints (3). Constraints (4) guarantee primary connectivity. Following Magnanti and Wolsey [4], a compact formulation providing the same linear programming bound can be obtained by replacing constraints (3) by a network flow subproblem. The formulation thus obtained is denoted by (AArbFlowr ): 1 1 2 2 (AArbFlowr ) Min (cij xij + cij xij ), (i,j )∈A
(1), (2), (4), (5), yjki −
j ∈V −{k}:(j,i)∈A
i∈V −{k}:(i,k)∈A
j ∈V :(i,j )∈A
k yik = 1,
yijk xij1 + xij2 , yijk 0,
yijk = 0,
∀k ∈ V − {r}, i ∈ V − {r, k},
∀k ∈ V − {r},
∀(i, j ) ∈ A, k ∈ V − {r} : i = k,
∀(i, j ) ∈ A, k ∈ V − {r} : i = k.
(6)
(7) (8) (9)
The compact formulation is attractive in the sense that, for medium sized problems, one can use available MIP packages to obtain the optimum value. In fact, this formulation is used to provide the results with up to 150 nodes given in Section 5 as well as the solutions for the example given in Fig. 2. For larger problems, one can follow Gouveia and Telhada and derive the following Lagrangean relaxation which is defined as follows: (i) relax the primary constraints and (ii) solve the Lagrangean subproblem which corresponds to a minimum cost arborescence problem (see [5] for more details).
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Fig. 2. Example of an instance that illustrates the non-symmetry of (AArbr ).
As noted before, a formulation is said to be symmetric if its linear programming relaxation optimum value does not change when the ordering of nodes is changed (see [8]). Changing the root node gives an example of reordering the nodes of the graph. It is easy to verify that (AArbr ) is not symmetric (see Fig. 2). In the upper part of Fig. 2 an instance of the MWST problem is depicted. Associated to each edge of that graph is a pair of costs (primary and secondary, respectively). In the lower left part of the same figure, the linear programming relaxation optimal solution of (AArbr ), for the case when r = 1, is presented. The linear programming relaxation optimal solution for the case when r = 6 is presented in the lower right part of Fig. 2. In both solutions, the values next to each arc indicate the values of the corresponding primary and secondary variables, xij1 and xij2 . Primary values are represented by double stroke lines whereas secondary ones are represented by simple lines. Note that the optimal values, for each of the two linear programming relaxations, differ. For the case when r = 1, that value is equal to 11.5, while for the case when r = 6 that value is equal to 15.8. This example shows that different root nodes may lead to different linear programming relaxation bounds. Unfortunately, there seems to be no apparent way of selecting a priori the root that yields the best bound. This observation gives the motivation for the reformulation technique presented in the next section. Before presenting the new formulation, we note for completeness that the directed multicommodity flow formulation presented by Balakrishnan et al. [7] (denoted by BMM in the remainder of the text) is symmetric and it differs from the compact version of the augmented formulation given before, only in the primary connectivity constraints (see below): (BMM)
(AArbFlowr ) yijk xij1 + xij2 , ∀(i, j ) ∈ A, k ∈ V − {r} 1 (V − {i, j }, {i})x 1 , ∀(i, j ) ∈ A : i ∈ X ij
S
yijk xij1 + xij2 , ∀(i, j ) ∈ A, k ∈ S yijk xij1 , ∀(i, j ) ∈ A, k ∈ P − {r}
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Gouveia and Telhada [5] have shown that v ∗ (BMML ) v ∗ (AArbkL ), for all k ∈ P . As noted before, computational experience reported in that same paper shows, however, that the difference between the two linear programming bounds appears to be quite small. Furthermore, for some instances, the CPU times involved in obtaining the linear programming bounds are better for the case of the (AArbFlowr ) formulation when compared with (BMM). 3. Reformulation by intersection The basic idea for the new formulation is to consider, at the same time, the previous formulation (AArbr ) for all possible values of r, and guarantee that the feasible solutions of the models associated to every root intersect each other, in the sense that a feasible solution for a model associated to a given root node r cannot use an arc which is not used (in the same direction or in the reversed direction) by the solution of the model associated to any other root. Let Ak (E) denote the set of arcs associated to root k. In order to guarantee that each directed solution is superimposed with the remaining ones, our model will use 1 and x 2 , to define the inclusion of edge {i, j }, respectively, with primary and additional undirected variables, x{ij } {ij } secondary equipment, in the solution. These variables are introduced in order to guarantee that each directed rooted solution matches with the undirected solution (or more precisely, a feasible undirected solution is a projection of a directed solution, for each possible root). Let { xij1k }(i,j )∈Ak (E),k∈P and { xij2k }(i,j )∈Ak (E),k∈P be the variables defining primary or secondary equipment, respectively, in the model whose root node is k. In order to guarantee that each directed solution matches with the same undirected solution, the following constraints must be included in the new model: 1 x{i,j ij1k + xj1ki , }=x 1 1k kj , x{k,j }=x
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P ,
2 ij2k + xj2ki , x{i,j }=x 2k 2 , kj x{k,j }=x
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P .
(10) (11) (12) (13)
The new formulation (denoted by SuperAArb) is completed by considering the constraints from all models (AArbk ) associated to the different root nodes k together with the previously given “intersection” constraints. Note that the objective function involves, now, the undirected variables, despite the fact that it could be written in terms of any of the directed variables set: 1 1 2 2 (SuperAArb) Min (c{i,j } x{i,j } + c{i,j } x{i,j } ), {i,j }∈E
1k (V − {j }, {j }) = 1, X
∀j ∈ P − {k}, k ∈ P ,
12k (V − {j }, {j }) = 1, X 12k (U )|U | − 1, X
∀j ∈ V − {k}, k ∈ P ,
(15)
∀U ⊆ V − {k}: |U |2, k ∈ P ,
(16)
1k (V − {i, j }, {i}) x1k , X ij 1 ij1k + xj1ki , x{i,j }=x 1k 1 kj , x{k,j }=x
2 2k kj , x{k,j }=x
∀(i, j ) ∈ Ak (E): i ∈ S, k ∈ P ,
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P ,
2 ij2k + xj2ki , x{i,j }=x
(14)
∀{i, j } ∈ E(V − {k}), k ∈ P ,
∀{i, j } ∈ ({k}), k ∈ P ,
(17) (10) (11) (12) (13)
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xij1k , xij2k 0,
∀(i, j ) ∈ Ak (E), k ∈ P ,
xij1 , xij2 ∈ {0, 1},
(18)
∀{i, j } ∈ E.
(19)
This new formulation has O(|E| + |P | × |A|) variables and O(|P | × (|P | + |V | + 2|V | + |A| + 2|E|)) constraints. The intersection constraints (10)–(13) guarantee that the projection of the set of feasible solutions of the (SuperAArb) linear programming relaxation onto the space defined by variables referring to a specific root satisfies all constraints included in the respective (AArbr ) model, that is: Proposition 1. PROJ{x 1
2 {i,j } ,x{i,j } }
{F (SuperAArbL )} ⊆ PROJ{x 1
2 {i,j } ,x{i,j } }
{F (AArbkL )}, for all k ∈ P .
As a consequence, the linear programming relaxation bound obtained by (SuperAArb) is at least as good as the bound obtained by any of the (AArbr ) linear programming relaxation. Proposition 2. v ∗ (SuperAArbL ) v ∗ (AArbkL ), for all k ∈ P . It remains to see what the relationship is between the linear programming relaxations of the two models, (SuperAArb) and (BMM) (which is the model with the best known linear programming bound, and already mentioned in the previous section). In all the results taken (see Section 5), the linear programming bounds of the two models are equal. However, as was shown in Telhada [9], we have that v ∗ (BMML ) v ∗ (SuperAArbL ) and, furthermore, there exist some instances for which a strict inequality holds. We skip the proof of this result as it is rather tedious. We note that it follows a similar proof given by Goemans and Myung [10]. In the same way that a network flow compact version of the formulation (AArbr ) was derived in [5], we can also obtain a compact version of the formulation (SuperAArb) by replacing the subtour elimination constraints by adequate flow constraints. However, as we shall show next, the intersection constraints allow us to use commodities defined only from a given primary source to every secondary node. To see this, first note that for each k, the directed subtour elimination constraints (16) are equivalent, under the presence of intersection constraints (10)–(13), to a unique set of undirected subtour elimination constraints. X12 (U )|U | − 1,
∀U ⊆ V : |U |2.
(20)
This result allows us to replace the directed subtour elimination constraints (16) by the more compact set of undirected subtour elimination constraints (20) in the formulation (SuperAArb). This modified formulation will be used as the starting point for deriving a Lagrangean relaxation scheme (see Section 4). For the purpose of providing the compact formulation, we shall additionally show that some of constraints (20) are redundant and thus, can be eliminated: 1 , x 2 , x 1k , x 2k } that satisfies (15), (10)–(13) and (18) also satisfies constraints (20) Proposition 3. A solution {x{i,j ij } {i,j } ij for subsets U that contain at least one primary node.
Proof. Consider a subset of nodes U that contains at least one primary node. Let k ∗ be a primary node in U . Using the intersection constraints (10)–(13) for node k ∗ , 12k ∗ (U − {i}, {i}). 12k ∗ (U ) = X X12 (U ) = X i∈U −{k ∗ }
Non-negativity constraints (18) guarantee that 12k ∗ (U − {i}, {i}) 12k ∗ (V − {i}, {i}). X X i∈U −{k ∗ }
i∈U −{k ∗ }
Using the indegree constraints (15) gives 12k ∗ (V − {i}, {i}) = X X 12 (U ) i∈U −{k ∗ }
i∈U −{k ∗ }
1 = |U | − 1.
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Fig. 3. Optimal solution for (SuperAArbL ) without the subtour elimination constraints.
The previous proposition shows that we need to consider subtour elimination constraints (20) in formulation (SuperAArb) only for subsets U that are contained in S. To highlight the need of using the remaining subtour elimination constraints to obtain a valid model for the problem, Fig. 3 shows the optimal solution produced by (SuperAArb) without the whole set of subtour elimination constraints for the same instance shown in Fig. 2. This solution is unfeasible, as it contains a subtour involving only nodes in S. Before obtaining the equivalent compact formulation, we note that by using the intersection constraints, for a given 12k ∗ (U ), and then by using the indegree constraints (15), we know primary node k ∗ , which guarantee that X 12 (U ) = X that the subtour elimination constraints (16) can be rewritten in equivalent cut form as follows: 12k ∗ (U C , U )1, X
∀U ⊆ S: |U |2 for some k ∗ ∈ P .
(21)
This observation shows that we can obtain a formulation with an equivalent linear programming bound by replacing the subtour elimination constraints (20) for sets U contained in S (or the cut constraints (21) for the same sets) in formulation (SuperAArb) by a restricted network flow system that only contains a commodity from any given primary node k ∗ to each secondary node, leading to the model (SuperAArbFlow): 1 2 2 1 (c{i,j (SuperAArbFlow) Min } x{i,j } + c{i,j } x{i,j } ), {i,j }∈E
(10).(15), (17).(19), yjki −
j ∈V −{i,k ∗ }
j ∈V −{i,k}
j ∈V −{k}
yjkk = 1, ∗
yijk = 0,
∀i ∈ V − {k ∗ , k}, k ∈ S,
∀k ∈ S, ∗
0 yijk xij1k + xij2k ,
(22) (23)
∗
∀(i, j ) ∈ Ak (E), k ∈ S.
(24)
As before, results obtained by using such a formulation will be reported in the Computational results section for medium sized instances. This compact version was used to solve the instance of Fig. 2 (which, in turn, was used to illustrate the lack of symmetry of the model (AArbk )). For this instance, the optimal solution of (SuperAArbL ) is integer and has value 17 (see Fig. 4) and this example can be used to show that the inequality in the statement of Proposition 2 can be strict. It is interesting to compare this formulation with the one we would have obtained by applying the intersection reformulation to the compact formulation (AArbFlowr ). Such a model would contain many more commodities which were shown to be redundant by using the intersection constraints and Proposition 3.
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Fig. 4. Optimal solution of (SuperAArbL ) for the instance shown in Fig. 2.
As will be shown in the section on computational results, the new formulation appears to produce very good linear programming bounds. However, the formulation becomes difficult to use for larger instances. In order to overcome the difficulty of solving its linear programming relaxations, we describe in the next section a Lagrangean relaxation based on formulation (SuperAArb). Note that, due to similar difficulties, Balakrishnan et al. [7] developed a dual ascent method based on the model (BMM) and have reported CPU times and gaps for instances up to 500 nodes and 5000 edges. The proportion of primary nodes on the set of instances tested by Balakrishnan et al. varies between 40% and 75%. 4. A Lagrangean relaxation We propose a Lagrangean relaxation scheme derived from the new formulation. As noted before, we use the model (SuperAArb), where constraints (16) were replaced by the simpler set, X 12 (U )|U | − 1,
∀U ⊆ V − {k}: |U |2, k ∈ P ,
(20)
and add the following redundant constraint, guaranteeing that |V | − 1 edges are included in the solution: X 12 (V ) = |V | − 1.
(25)
We observe that the subproblem defined on the undirected variables can be viewed as an undirected spanning tree model. Note that the set of constraints (20) could be further reduced as explained in the previous section. However, we decided to maintain the whole set for the following two reasons: (i) we do not need to check whether the problem of selecting |V | − 1 edges with minimum cost and not containing subtours involving secondary nodes is polynomial and (ii) most importantly the relaxed problem becomes stronger if constraints (20) for all subsets are maintained in the subproblem. We can now derive a Lagrangean relaxation by dualizing the primary connectivity constraints (17) and the intersection 2k constraints (10)–(13), attaching multipliers kij to the first set of constraints and attaching 1k {i,j } and {i,j } to the second set of constraints. The relaxed problem obtained is as follows: ⎛ ⎞ 1 1 2 2 (SuperAArbLR) Min (c{i,j kij ⎝xij1k − xli1k ⎠ } x{i,j } + c{i,j } x{i,j } ) + {i,j }∈E
+
k∈P {i,j }∈E
k∈P (i,j )∈Ak (E):i∈S 1 2k 2 [1k xij1k + xj1ki − x{i,j xij2k + xj2ki − x{i,j } ) + {i,j } ( } )], {i,j } (
l∈V −{i,j }
L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599 – 3611
X 12 (V ) = |V | − 1,
(25)
1k (V − {j }, {j }) = 1, X
∀j ∈ P − {k}, k ∈ P ,
12k (V − {j }, {j }) = 1, X X (U )|U | − 1,
(14)
∀j ∈ V − {k}, k ∈ P ,
(15)
∀U ⊆ V − {k}: |U |2, k ∈ P ,
12
xij1k , xij2k 0,
(20)
∀(i, j ) ∈ Ak (E), k ∈ P ,
xij1 , xij2 ∈ {0, 1},
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(18)
∀{i, j } ∈ E.
(5)
For any given value of the multipliers, the relaxed problem can be separated into |P | + 1 subproblems: (i) an undirected spanning tree problem involving the undirected variables and which can be easily solved by the algorithm given by Prim [11], for example, and (ii) |P | semi-assignment problems which can be easily solved by inspection, one for each root node k: 1 2 2 (SuperAArbLRU ) Min (c1{i,j } x{i,j } + c{i,j } x{i,j } ), {i,j }∈E
X12 (V ) = |V | − 1,
(25)
X 12 (U )|U | − 1, xij1 , xij2 ∈ {0, 1},
∀U ⊆ V − {k}: |U |2, k ∈ P ,
(20)
∀{i, j } ∈ E,
(SuperAArbLRk ),
k∈P
Min
(5)
1k 1k 1k 2k 2k 2k 2k (c1k ij xij + cj i xj i + cij xij + cj i xj i ),
{i,j }∈Ak (E)
1k (V − {j }, {j }) = 1, X 12k
X
∀j ∈ P − {k}, k ∈ P ,
(V − {j }, {j }) = 1,
xij1k , xij2k 0,
(14)
∀j ∈ V − {k}, k ∈ P ,
(15)
∀(i, j ) ∈ Ak (E), k ∈ P .
(18)
The modified costs used in these subproblems are defined as: 1 c1{i,j } = c{i,j 1k ∀{i, j } ∈ E, }− {i,j } , k∈P 2 c2{i,j } = c{i,j }−
2k {i,j } ,
∀{i, j } ∈ E,
k∈P 1k c1k ij = {i,j } ,
c1j i = 1k {i,j } , = 1k kli , {i,j } −
∀{i, j } ∈ E : i, j ∈ P ,
c1k ij
k c1j i = 1k {i,j } + j i ,
l∈V −{i,j }
k 1k c1k ij = {i,j } + ij , k 1k c1k ij = {i,j } + ij −
c1j i = 1k {i,j } − l∈V −{i,j }
kli ,
l∈V −{i,j }
klj ,
∀{i, j } ∈ E : i ∈ S, j ∈ P , ∀{i, j } ∈ E : i ∈ P , j ∈ S,
k c1j i = 1k {i,j } + j i −
l∈V −{i,j }
klj ,
∀{i, j } ∈ E : i, j ∈ S.
To find an approximation of the optimal multipliers that produce the best lower bound value we used the method 2k proposed by Camerini et al. [12]. The multipliers = {kij , 1k {i,j } , {i,j } } are updated at each step as follows: k+1 = k + tk d k . The step length as well as the improvement directions is updated according to the Camerini–Fratta–Maffioli rule.
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At each step of the multipliers approximation procedure, we compute a feasible solution for the problem. The cost of this solution gives an upper bound on the value of the optimal solution and is obtained by examining the relaxed minimum spanning tree solution and changing (when needed) to primary the costs of the edges in paths connecting primary nodes. 5. Computational results We report our computational experience on the model and methods proposed in this paper. For our experiments, we used Euclidean cost instances and randomly generated instances (see [5] for details on those instances construction procedure). The number of nodes of the instances used varies between 50, 75, 100, 125 and 150. For each instance, we considered 25%, 50% and 75% of the nodes to be primary. For each set of parameters (number of nodes and proportion of primary nodes), 5 Euclidean instances and 5 random instances were generated. The instances with 50 nodes are the instances already used in our previous work [5]. We note, however, that the results reported here regarding the (AArb) model may be better than the ones reported earlier. This is due to some improvements on both the model and the machine used. For every generated instance, we use some of the preprocessing tests proposed by Balakrishnan et al. [3]. The computational times for these tests were negligible for the instances tested. All instances reported in this table have a 25% density. For a better understanding of the reported results, instances are divided according to the proportion of primary nodes and to the generation method. The results reported in Table 1 show the impact of the preprocessing tests. The first three columns describe each class of instances. In the first column, number of nodes (primary + secondary) and number of edges are given. The number of primary nodes is also given in percentage in the second column. The third column indicates the type of edge cost generation method used, either random or Euclidean. The last two columns give similar information to the first two columns, after reduction tests have been performed. For example, the first row gives information on instances with 50 nodes, 12 primary (which corresponds to 25% of all nodes of that instance) and 38 secondary, and 306 edges generated by the Euclidean procedure. After performing the reduction tests, instances ended up with an average of 47.8 nodes (9.8 primary and 38 secondary) and with an average of 119.6 edges. The proportion of primary nodes, after performing the reduction tests, is equal to 20%. First of all, one observes that, in almost all cases, the number of edges is significantly reduced and that the higher the percentage of primary nodes is, the higher is the reduction effect. Another interesting fact is that the reduction tests are more effective for Euclidean instances rather than random instances. However, this happens especially when the number of nodes increases. In Table 2, computational results are reported for the 3 compact formulations referred to in this paper. We used version 9 of the CPLEX MIP solver for all 3 formulations and we report the CPU time spent, in seconds, until the LP relaxation bound was obtained (and the corresponding gap), and also the CPU time until the integer optimum was obtained. All tests were performed on a Pentium 4 CPU 3.20 GHz, 512 MB of RAM machine. Instances considered have now densities varying between 10%, 25% and 40%. To observe both the effect of the number of nodes of the density on the performance of the three models, classes of instances were separated according to those parameters. The first column indicates the number of nodes and the second column indicates the density, respectively, for the group of instances considered in each line. For each model ((AArb), (SuperAArb) and (BMM)), we report CPU times to reach the linear programming relaxation optimum and the integer optimum, as well as the LP gap thus obtained. The results reported in Table 2 show that the linear programming bound of the formulation (SuperAArb) is very tight (for all the cases reported, gaps are equal to zero) and that it improves considerably on the linear programming bound of the formulation (AArb). However, the improved bounds have the same quality as the bounds given by the linear programming relaxation of the (BMM) formulation. Additionally, (BMM) takes less CPU time to obtain the linear programming bounds as well as the integer optimum values. Instances with 150 nodes and density 40% could not be solved by the (AArb) model in less than 10,000 s. Table 3 provides similar information, but now we give the results in terms of the proportion of primary nodes of the instances. The conclusions are similar to the conclusions taken from Table 2. It is interesting to point out that for instances with a larger number of primary nodes, the CPU times of the two best models are nearly identical. Unfortunately, for cases with a small number of primary nodes, the CPU times of the new model are substantially
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Table 1 Size of the instances after preprocessing Original
Reduced
N (P + S)/M
%P (%)
50(12 + 38)/306
25
50(25 + 25)/306
50
50(37 + 13)/306
75
75(18 + 57)/693
25
75(37 + 38)/693
50
75(56 + 19)/693
75
100(25 + 75)/1237
25
100(50 + 50)/1237
50
100(75 + 25)/1237
75
125(31 + 94)/1937
25
125(62 + 63)/1937
50
125(93 + 32)/1937
75
150(37 + 113)/2793
25
150(75 + 75)/2793
50
150(112 + 38)/2793
75
N(P + S)/M
%P (%)
Euclidean Random Euclidean Random Euclidean Random
47.8(9.8 + 38)/119.6 47.8(9.8 + 38)/107.8 36.2(11.2 + 25)/84 37.4(12.4 + 25)/156.8 24(11 + 13)/44 22.2(9.2 + 13)/60.8
20 20 31 33 46 40
Euclidean Random Euclidean Random Euclidean Random
70.8(13.8 + 57)/210 69.4(12.4 + 57)/507.2 56.2(18.2 + 38)/157.8 58.8(20.8 + 38)/334.2 34.2(15.2 + 19)/65.6 31.4(12.4 + 19)/77.6
19 18 32 35 44 38
Euclidean Random Euclidean Random Euclidean Random
82.4(7.4 + 75)/589.2 92.4(17.4 + 75)/257 55(5 + 50)/234.6 74(24 + 50)/176 28(3 + 25)/72.4 44.2(19.2 + 25)/75.2
9 19 9 32 11 43
Euclidean Random Euclidean Random Euclidean Random
100.4(6.4 + 94)/810.4 118(24 + 94)/316.2 68.4(5.4 + 63)/337.4 92.2(29.2 + 63)/229.4 35.2(3.2 + 32)/107.8 56(24 + 32)/101
6 20 8 32 9 42
Euclidean Random Euclidean Random Euclidean Random
122.2(9.2 + 113)/1170.8 139(26 + 113)/413.2 81.8(6.8 + 75)/460 112.8(37.8 + 75)/292 40.6(2.6 + 38)/127.4 68.6(30.6 + 38)/127.2
7 19 8 33 6 44
larger than the CPU time provided by the (BMM) model. This can be explained by the fact that, for the instances with a smaller number of primary nodes, the number of primary connectivity constraints (17) of the (SuperAArb) model, taken for each arc incident on a secondary node, is considerably higher than the number of flow linking constraints of the (BMM) model. More interesting for the new model is the information given in Table 4 that shows the results in terms of the two different methods of generating the costs for the instances. In this case, the CPU times given by the new formulation (SuperAArb) are better than the CPU times given by the (BMM) formulation for the Euclidean instances but much worse for random instances. The CPU times reported for the new model indicate that the CPU times may become critical when solving larger instances. For such cases, we suggest using, as an alternative, the Lagrangean relaxation scheme proposed in Section 4. Therefore, we also tested that procedure in order to evaluate its behaviour for instances that cannot be easily solved with the linear programming relaxation. Table 5 shows some encouraging results for 5 Euclidean instances and 5 random instances with 500 nodes and 5000 edges. For each instance, the percentage of primary nodes was considered to be equal to, respectively, 25%, 50% and 75%. Average number of iterations, average duration and average gap are again reported. In many cases, integral optimality was obtained, although in some cases a large number of iterations was needed to obtain such bounds. Nevertheless, the reported CPU times appear to be quite acceptable and are lower for Euclidean instances.
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Table 2 Results given by the three models according to number of nodes N
D (%)
AArb
SuperAArb
LP Gap (%)
CPU (s)
Int.
LP
CPU (s)
Gap (%)
BMM Int.
LP
CPU (s)
CPU (s)
Gap (%)
Int. CPU (s)
CPU (s)
50
10 25 40
0.12 0.19 0.42
0.56 0.59 1.50
0.76 1.26 4.31
0.00 0.00 0.00
0.75 0.76 1.42
0.84 0.87 1.56
0.00 0.00 0.00
0.54 0.52 1.34
0.61 0.61 1.44
75
10 25 40
0.02 0.09 0.33
4.93 8.30 14.79
5.57 42.91 204.77
0.00 0.00 0.00
6.47 16.04 15.39
6.79 18.91 16.07
0.00 0.00 0.00
5.47 9.40 10.21
5.77 15.44 10.65
100
10 25 40
0.13 0.24 0.35
16.99 8.17 40.74
187.51 26.44 230.51
0.00 0.00 0.00
45.99 25.44 77.04
46.61 26.16 79.05
0.00 0.00 0.00
19.53 9.27 30.14
20.09 10.11 31.03
125
10 25 40
0.01 0.13 0.32
50.35 35.98 75.74
96.17 76.36 3907.71
0.00 0.00 0.00
76.61 93.10 274.70
79.72 94.49 366.66
0.00 0.00 0.00
27.55 14.84 68.12
28.97 16.05 71.28
150
10 25 40
0.00 0.17 –
51.99 42.04 –
61.59 187.42 –
0.00 0.00 0.06
255.36 353.96 1604.87
257.74 475.08 3926.62
0.00 0.00 0.00
61.10 56.50 994.13
62.92 59.92 1259.65
Table 3 Results given by the three models according to the proportion of primary nodes %P (%)
AArb
SuperAArb
LP
25 50 75
Int.
LP
BMM Int.
LP
Int.
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
0.38 0.09 0.04
55.99 12.58 0.33
945.67 26.94 0.42
0.00 0.00 0.00
202.87 39.26 0.35
245.57 40.45 0.44
0.00 0.00 0.00
49.26 12.50 0.32
52.73 13.12 0.40
Table 4 Results given by the three models according to cost generation method G
AArb
SuperAArb
LP
Euclidean Random
Int.
LP
BMM Int.
LP
Int.
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
Gap (%)
CPU (s)
CPU (s)
0.22 0.14
15.09 33.77
61.98 622.24
0.00 0.00
12.13 157.97
13.74 187.35
0.00 0.00
14.86 29.21
16.00 31.03
Table 5 Some results obtained for instances with 500 nodes N/M
500/5000
%P (%)
25 50 75
Euclidean
Random
Iter.
CPU (s)
Gap (%)
Iter.
CPU (s)
Gap (%)
3535 1186.6 2383.4
6353.8 464 116.2
0.078 0.000 0.002
8554.2 4491 327
7596.6 2634.6 59
0.049 0.084 0.000
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6. Conclusions Previously, Gouveia and Telhada [5] proposed a formulation for the MWST problem that is not symmetric (that is, its linear programming relaxation value depends on the choice of the node to be the root of the tree). In this paper we proposed a new formulation whose linear programming relaxation bound is at least as good as the linear programming relaxation bound obtained by using the previous formulation for all possible root selections. The basic idea for obtaining the new formulation is to consider, at the same time, the previous formulation for all possible root choices, and guarantee that the feasible solutions of the models associated to every root intersect each other. Our results show that the linear programming relaxation of this so-called “reformulation by intersection” substantially improves on the linear programming relaxation of the original model. The new lower bounds are comparable with the lower bounds of the best known formulation for the problem. We also suggested a new Lagrangean relaxation based on the new model and presented results with up to 500 nodes and 5000 edges. Reported gaps were less than 1% and are equal to zero in many cases. We conclude this paper by noting that the reformulation by intersection idea can be applied to improve the linear programming relaxation of formulations of many other discrete optimization problems, provided that the original given formulation “suffers” from lack of symmetry. Acknowledgement This work was supported by contract grant sponsor: FCT under contract Grant number: Po CTI-ISFL-1-152. References [1] Current JR, Revelle CS, Cohon JL. The hierarchical network design problem. European Journal of Operational Research 1986;27:57–66. [2] Duin C, Volgenant T. The multi-weighted Steiner tree problem. Annals of Operations Research 1991;33:451–69. [3] Balakrishnan A, Magnanti T, Mirchandani P. Modeling and heuristic worst-case performance analysis of the two-level network design problem. Management Science 1994;40(7):846–67. [4] Magnanti T, Wolsey L. Optimal trees. Network models. Handbooks in operations research and management science, vol. 7. 1995. p. 503–615. [5] Gouveia L, Telhada J. An augmented arborescence formulation for the two-level network design problem. Annals of Operations Research 2001;106(1):47–61. [6] Khoury BN, Pardalos PM, Hearn DW. Equivalent formulations for the Steiner problem in graphs. In: Du D-Z, Pardalos PM, editors. Network optimization problems. 1993. p. 111–23. [7] Balakrishnan A, Magnanti T, Mirchandani P. Dual-based algorithm for multi-level network design. Management Science 1994;40(5):567–81. [8] Yannakakis M. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 1991;43(3): 441–66. [9] J. Telhada, Desenho Topológico de Redes Hierárquicas. PhD thesis, University of Lisbon; 2004. [10] Goemans MX, Myung Y-S. A catalog of Steiner tree formulations. Networks 1993;23:19–28. [11] Prim RC. Shortest connection networks and some generalizations. Bell System Technical Journal 1957;36:1389–401. [12] Camerini PM, Fratta L, Maffioli F. On improving relaxation methods by modified gradient techniques. Mathematical Programming Study 1975;3:26–34.