Computers & Operations Research 35 (2008) 3599–3611www.elsevier.com/locate/cor

The multi-weighted Steiner tree problem: A reformulationby intersection

Luis Gouveia∗, João TelhadaDEIO-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 apreviously proposed formulation for the problem is non-symmetric in the sense that the corresponding linear programming relaxationbounds depend on the node selected as a root of the tree. The new formulation (the reformulation by intersection) is obtained byintersecting 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 programmingrelaxation of each of the rooted formulations and is comparable with the linear programming bounds of the best formulation knownfor the problem. A Lagrangean relaxation scheme derived from the new formulation is also proposed and tested, with quite favourableresults, on instances with up to 500 nodes and 5000 edges.� 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 theset of primary nodes and S is the set of secondary nodes. To each edge e ∈ E, we associate two positive costs, c1

e

and c2e , 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 andsecondary links. Primary links are more expensive than secondary links, that is, for each e ∈ E we have c1

e > c2e . The

objective is to find a minimum cost connected network such that each pair of primary nodes is connected by a path withprimary links. Additionally, any two secondary nodes, or a secondary node and a primary node, can be connected bya path using either primary or secondary links. Fig. 1 depicts an example of such a two-level spanning tree with threeprimary 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 byDuin and Volgenant [2], where some reduction tests were proposed. More recently, the relationship between alternativemodels (e.g., directed and undirected models) was examined in Balakrishnan et al. [3], where it was pointed out thatwhen 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: legouveia@fc.ul.pt (L. Gouveia), jmtelhada@fc.ul.pt (J. Telhada).

0305-0548/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2007.03.003

3600 L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611

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 onthe subset P of primary nodes (see Fig. 1). Thus, a formulation for the MWST problem can be obtained by combininga formulation for the minimum cost Steiner tree problem together with a formulation for the minimum cost spanningtree problem (see [4] for formulations on both problems).

Previously, Gouveia and Telhada [5] proposed a formulation for the MWST problem which contains constraintsfrom a Steiner tree formulation by Khoury et al. [6] to guarantee primary connectivity (that is, to guarantee that allprimary nodes are connected in a primary level network). This formulation can be seen as an arborescence formulationaugmented with such primary connectivity constraints. Gouveia and Telhada have shown that the lower bounds givenby the linear programming relaxation of the new formulation are not better than the lower bounds given by thestrongest linear programming relaxation proposed by Balakrishnan et al. [7] (see [5] for some results). However, thecomputational results given by Gouveia and Telhada show that in many instances the difference between the two valuesis quite small. Furthermore, for several instances, CPU times needed to solve the linear programming relaxation of thenew formulation were smaller than the CPU times needed to obtain the linear programming relaxation optimum valueof the Balakrishnan et al. [7] model.

The fact that the new formulation is not symmetric (that is, its linear programming relaxation value depends on thechoice of the node to be the root of the tree) gives the theoretical motivation for this work. More precisely, it motivatesa new formulation whose linear programming relaxation bound is at least as good as the linear programming relaxationbound 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 importantnotation along with relevant results to the following sections. A Lagrangean relaxation scheme derived from the newformulation 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 withonly one end node in a subset of nodes U is denoted by �(U). In particular, when U is a singleton, �({k}) representsthe set of edges incident in node k. If (P) is a valid formulation for the problem, (PL) denotes the correspondinglinear programming relaxation obtained by replacing the binary constraints by constraints stating that each variablevaries 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 mixedinteger programming problem. If (P) is defined on an extended set of variables (x, y), then the projection of its feasiblesolutions 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 boundsobtained by using the corresponding linear programming relaxations. See, for example, Magnanti and Wolsey [4] ondirected models for the minimum spanning tree problem. In this case, we consider an arc set A which contains two arcs

L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611 3601

(i, j) and (j, i) for each edge {i, j} in E with the same cost as the original edge. Directing the problem also requireschoosing a root node. In the context of the MWST problem, the root must always be a primary node and will usuallybe 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 directedMWST involves characterizing the set of arcs where primary equipment is to be installed and the set of arcs wheresecondary equipment is to be installed (see [5,7] for more details). Therefore, we shall use the two sets of variables,x1ij and x2

ij , which indicate whether arc (i, j) is being used in the directed version of the problem, with primary orsecondary equipment, respectively.

To simplify notation used in the remainder of the text, �Xk(U) = ∑(i,j)∈A:i,j∈U xk

ij , k = 1, 2, and X12(U) =∑(i,j)∈A:i,j∈U (x1

ij + x2ij ). For each cutset induced by a node subset U , �Xk(UC, U) and �X12(UC, U) follow the

same pattern, where UC 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:

(AArbr ) Min∑

(i,j)∈A

(c1ij x

1ij + c2

ij x2ij ),

�X1(V − {j}, {j}) = 1, ∀j ∈ P − {r}, (1)

�X12(V − {j}, {j}) = 1, ∀j ∈ S, (2)

�X12(U)� |U | − 1, ∀U ⊆ V − {r} : |U |�2, (3)

�X1(V − {i, j}, {i})�x1ij , ∀(i, j) ∈ A : i ∈ S, (4)

x1ij , x

2ij ∈ {0, 1}, ∀(i, j) ∈ A. (5)

This formulation has o(2|A|) variables and o(|V | + 2|V | + |A|) constraints. Connectivity is guaranteed by theindegree constraints (1) and (2) together with the subtour elimination constraints (3). Constraints (4) guarantee primaryconnectivity.

Following Magnanti and Wolsey [4], a compact formulation providing the same linear programming bound canbe obtained by replacing constraints (3) by a network flow subproblem. The formulation thus obtained is denoted by(AArbFlowr ):

(AArbFlowr ) Min∑

(i,j)∈A

(c1ij x

1ij + c2

ij x2ij ),

(1), (2), (4), (5),∑

j∈V −{k}:(j,i)∈A

ykji −

∑j∈V :(i,j)∈A

ykij = 0, ∀k ∈ V − {r}, i ∈ V − {r, k}, (6)

∑i∈V −{k}:(i,k)∈A

ykik = 1, ∀k ∈ V − {r}, (7)

ykij �x1

ij + x2ij , ∀(i, j) ∈ A, k ∈ V − {r} : i �= k, (8)

ykij �0, ∀(i, j) ∈ A, k ∈ V − {r} : i �= k. (9)

The compact formulation is attractive in the sense that, for medium sized problems, one can use available MIPpackages to obtain the optimum value. In fact, this formulation is used to provide the results with up to 150 nodes givenin Section 5 as well as the solutions for the example given in Fig. 2. For larger problems, one can follow Gouveia andTelhada and derive the following Lagrangean relaxation which is defined as follows: (i) relax the primary constraintsand (ii) solve the Lagrangean subproblem which corresponds to a minimum cost arborescence problem (see [5] formore details).

3602 L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611

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 notchange when the ordering of nodes is changed (see [8]). Changing the root node gives an example of reordering thenodes 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 isa pair of costs (primary and secondary, respectively). In the lower left part of the same figure, the linear programmingrelaxation optimal solution of (AArbr ), for the case when r = 1, is presented. The linear programming relaxationoptimal solution for the case when r = 6 is presented in the lower right part of Fig. 2. In both solutions, the valuesnext to each arc indicate the values of the corresponding primary and secondary variables, x1

ij and x2ij . Primary values

are represented by double stroke lines whereas secondary ones are represented by simple lines. Note that the optimalvalues, 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. Unfor-tunately, there seems to be no apparent way of selecting a priori the root that yields the best bound. This observationgives 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 formulationpresented by Balakrishnan et al. [7] (denoted by BMM in the remainder of the text) is symmetric and it differs from thecompact version of the augmented formulation given before, only in the primary connectivity constraints (see below):

(AArbFlowr ) (BMM)

ykij �x1

ij + x2ij , ∀(i, j) ∈ A, k ∈ V − {r} yk

ij �x1ij + x2

ij , ∀(i, j) ∈ A, k ∈ S

�X1(V − {i, j}, {i})�x1ij , ∀(i, j) ∈ A : i ∈ S yk

ij �x1ij , ∀(i, j) ∈ A, k ∈ P − {r}

L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611 3603

Gouveia and Telhada [5] have shown that v∗(BMML)�v∗(AArbkL), for all k ∈ P . As noted before, computationalexperience reported in that same paper shows, however, that the difference between the two linear programming boundsappears to be quite small. Furthermore, for some instances, the CPU times involved in obtaining the linear programmingbounds 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 allpossible values of r , and guarantee that the feasible solutions of the models associated to every root intersect eachother, in the sense that a feasible solution for a model associated to a given root node r cannot use an arc which is notused (in the same direction or in the reversed direction) by the solution of the model associated to any other root. LetAk(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 useadditional undirected variables, x1{ij} and x2{ij}, to define the inclusion of edge {i, j}, respectively, with primary andsecondary equipment, in the solution. These variables are introduced in order to guarantee that each directed rootedsolution matches with the undirected solution (or more precisely, a feasible undirected solution is a projection of adirected solution, for each possible root). Let {�x1k

ij }(i,j)∈Ak(E),k∈P and {�x2kij }(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:

x1{i,j} = �x1kij + �x1k

ji , ∀{i, j} ∈ E(V − {k}), k ∈ P, (10)

x1{k,j} = �x1kkj , ∀{i, j} ∈ �({k}), k ∈ P, (11)

x2{i,j} = �x2kij + �x2k

ji , ∀{i, j} ∈ E(V − {k}), k ∈ P, (12)

x2{k,j} = �x2kkj , ∀{i, j} ∈ �({k}), k ∈ P. (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 theobjective function involves, now, the undirected variables, despite the fact that it could be written in terms of any ofthe directed variables set:

(SuperAArb) Min∑

{i,j}∈E

(c1{i,j}x1{i,j} + c2{i,j}x2{i,j}),

�X1k(V − {j}, {j}) = 1, ∀j ∈ P − {k}, k ∈ P, (14)

�X12k(V − {j}, {j}) = 1, ∀j ∈ V − {k}, k ∈ P, (15)

�X12k(U)� |U | − 1, ∀U ⊆ V − {k}: |U |�2, k ∈ P, (16)

�X1k(V − {i, j}, {i})� �x1kij , ∀(i, j) ∈ Ak(E): i ∈ S, k ∈ P, (17)

x1{i,j} = �x1kij + �x1k

ji , ∀{i, j} ∈ E(V − {k}), k ∈ P, (10)

x1{k,j} = �x1kkj , ∀{i, j} ∈ �({k}), k ∈ P, (11)

x2{i,j} = �x2kij + �x2k

ji , ∀{i, j} ∈ E(V − {k}), k ∈ P, (12)

x2{k,j} = �x2kkj , ∀{i, j} ∈ �({k}), k ∈ P, (13)

3604 L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611

�x1kij , �x2k

ij �0, ∀(i, j) ∈ Ak(E), k ∈ P, (18)

x1ij , x

2ij ∈ {0, 1}, ∀{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 constraintsincluded in the respective (AArbr ) model, that is:

Proposition 1. PROJ{x1{i,j},x2{i,j}}{F(SuperAArbL)} ⊆ PROJ{x1{i,j},x2{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 thebound 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 previoussection). 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 instancesfor which a strict inequality holds. We skip the proof of this result as it is rather tedious. We note that it follows a similarproof 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 alsoobtain a compact version of the formulation (SuperAArb) by replacing the subtour elimination constraints by adequateflow constraints. However, as we shall show next, the intersection constraints allow us to use commodities definedonly from a given primary source to every secondary node. To see this, first note that for each k, the directed subtourelimination constraints (16) are equivalent, under the presence of intersection constraints (10)–(13), to a unique set ofundirected 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 undirectedsubtour elimination constraints (20) in the formulation (SuperAArb). This modified formulation will be used as thestarting point for deriving a Lagrangean relaxation scheme (see Section 4). For the purpose of providing the compactformulation, we shall additionally show that some of constraints (20) are redundant and thus, can be eliminated:

Proposition 3. A solution {x1{i,j}, x2{i,j}, x1kij , x2k

ij } that satisfies (15), (10)–(13) and (18) also satisfies constraints (20)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 . Usingthe intersection constraints (10)–(13) for node k∗,

X12(U) = �X12k∗(U) =

∑i∈U−{k∗}

�X12k∗(U − {i}, {i}).

Non-negativity constraints (18) guarantee that∑

i∈U−{k∗}�X12k∗

(U − {i}, {i})�∑

i∈U−{k∗}�X12k∗

(V − {i}, {i}).

Using the indegree constraints (15) gives

X12(U)�∑

i∈U−{k∗}�X12k∗

(V − {i}, {i}) =∑

i∈U−{k∗}1 = |U | − 1. �

L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611 3605

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 eliminationconstraints to obtain a valid model for the problem, Fig. 3 shows the optimal solution produced by (SuperAArb) withoutthe whole set of subtour elimination constraints for the same instance shown in Fig. 2. This solution is unfeasible, asit contains a subtour involving only nodes in S.

Before obtaining the equivalent compact formulation, we note that by using the intersection constraints, for a givenprimary node k∗, which guarantee that X12(U) = �X12k∗

(U), and then by using the indegree constraints (15), we knowthat the subtour elimination constraints (16) can be rewritten in equivalent cut form as follows:

�X12k∗(UC, U)�1, ∀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 replacingthe subtour elimination constraints (20) for sets U contained in S (or the cut constraints (21) for the same sets) informulation (SuperAArb) by a restricted network flow system that only contains a commodity from any given primarynode k∗ to each secondary node, leading to the model (SuperAArbFlow):

(SuperAArbFlow) Min∑

{i,j}∈E

(c1{i,j}x1{i,j} + c2{i,j}x2{i,j}),

(10).(15), (17).(19),∑

j∈V −{i,k}ykji −

∑j∈V −{i,k∗}

ykij = 0, ∀i ∈ V − {k∗, k}, k ∈ S, (22)

∑j∈V −{k}

ykjk = 1, ∀k ∈ S, (23)

0�ykij � �x1k∗

ij + �x2k∗ij , ∀(i, j) ∈ Ak∗

(E), k ∈ S. (24)

As before, results obtained by using such a formulation will be reported in the Computational results section formedium sized instances. This compact version was used to solve the instance of Fig. 2 (which, in turn, was used toillustrate the lack of symmetry of the model (AArbk)). For this instance, the optimal solution of (SuperAArbL) isinteger and has value 17 (see Fig. 4) and this example can be used to show that the inequality in the statement ofProposition 2 can be strict.

It is interesting to compare this formulation with the one we would have obtained by applying the intersectionreformulation to the compact formulation (AArbFlowr ). Such a model would contain many more commodities whichwere shown to be redundant by using the intersection constraints and Proposition 3.

3606 L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611

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 linearprogramming bounds. However, the formulation becomes difficult to use for larger instances. In order to overcome thedifficulty of solving its linear programming relaxations, we describe in the next section a Lagrangean relaxation basedon 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 primarynodes 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,

X12(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:

X12(V ) = |V | − 1. (25)

We observe that the subproblem defined on the undirected variables can be viewed as an undirected spanning treemodel. 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 problemof selecting |V | − 1 edges with minimum cost and not containing subtours involving secondary nodes is polynomialand (ii) most importantly the relaxed problem becomes stronger if constraints (20) for all subsets are maintained in thesubproblem.

We can now derive a Lagrangean relaxation by dualizing the primary connectivity constraints (17) and the intersectionconstraints (10)–(13), attaching multipliers �k

ij to the first set of constraints and attaching �1k{i,j} and �2k

{i,j} to the secondset of constraints. The relaxed problem obtained is as follows:

(SuperAArbLR) Min∑

{i,j}∈E

(c1{i,j}x1{i,j} + c2{i,j}x2{i,j}) +∑k∈P

∑

(i,j)∈Ak(E):i∈S

�kij

⎛⎝�x1k

ij −∑

l∈V −{i,j}�x1kli

⎞⎠

+∑k∈P

∑{i,j}∈E

[�1k{i,j}(�x1k

ij + �x1kji − x1{i,j}) + �2k

{i,j}(�x2kij + �x2k

ji − x2{i,j})],

L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611 3607

X12(V ) = |V | − 1, (25)

�X1k(V − {j}, {j}) = 1, ∀j ∈ P − {k}, k ∈ P, (14)

�X12k(V − {j}, {j}) = 1, ∀j ∈ V − {k}, k ∈ P, (15)

X12(U)� |U | − 1, ∀U ⊆ V − {k}: |U |�2, k ∈ P , (20)

�x1kij , �x2k

ij �0, ∀(i, j) ∈ Ak(E), k ∈ P , (18)

x1ij , x

2ij ∈ {0, 1}, ∀{i, j} ∈ E. (5)

For any given value of the multipliers, the relaxed problem can be separated into |P | + 1 subproblems: (i) anundirected spanning tree problem involving the undirected variables and which can be easily solved by the algorithmgiven by Prim [11], for example, and (ii) |P | semi-assignment problems which can be easily solved by inspection, onefor each root node k:

(SuperAArbLRU) Min∑

{i,j}∈E

(c1{i,j}x1{i,j} + c2{i,j}x2{i,j}),

X12(V ) = |V | − 1, (25)

X12(U)� |U | − 1, ∀U ⊆ V − {k}: |U |�2, k ∈ P , (20)

x1ij , x

2ij ∈ {0, 1}, ∀{i, j} ∈ E, (5)

(SuperAArbLRk), k ∈ P Min∑

{i,j}∈Ak(E)

(c1kij x1k

ij + c1kji x

1kji + c2k

ij x2kij + c2k

ji x2kji ),

�X1k(V − {j}, {j}) = 1, ∀j ∈ P − {k}, k ∈ P , (14)

�X12k(V − {j}, {j}) = 1, ∀j ∈ V − {k}, k ∈ P, (15)

�x1kij , �x2k

ij �0, ∀(i, j) ∈ Ak(E), k ∈ P . (18)

The modified costs used in these subproblems are defined as:

c1{i,j} = c1{i,j} −∑k∈P

�1k{i,j}, ∀{i, j} ∈ E,

c2{i,j} = c2{i,j} −∑k∈P

�2k{i,j}, ∀{i, j} ∈ E,

c1kij = �1k

{i,j}, c1ji = �1k

{i,j}, ∀{i, j} ∈ E : i, j ∈ P ,

c1kij = �1k

{i,j} −∑

l∈V −{i,j}�kli , c1

ji = �1k{i,j} + �k

ji , ∀{i, j} ∈ E : i ∈ S, j ∈ P ,

c1kij = �1k

{i,j} + �kij , c1

ji = �1k{i,j} −

∑l∈V −{i,j}

�klj , ∀{i, j} ∈ E : i ∈ P, j ∈ S,

c1kij = �1k

{i,j} + �kij −

∑l∈V −{i,j}

�kli , c1

ji = �1k{i,j} + �k

ji −∑

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 methodproposed by Camerini et al. [12]. The multipliers � = {�k

ij , �1k{i,j}, �

2k{i,j}} are updated at each step as follows:

�k+1 = �k + tkdk .

The step length as well as the improvement directions is updated according to the Camerini–Fratta–Maffioli rule.

3608 L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611

At each step of the multipliers approximation procedure, we compute a feasible solution for the problem. The costof this solution gives an upper bound on the value of the optimal solution and is obtained by examining the relaxedminimum spanning tree solution and changing (when needed) to primary the costs of the edges in paths connectingprimary nodes.

5. Computational results

We report our computational experience on the model and methods proposed in this paper. For our experiments, weused Euclidean cost instances and randomly generated instances (see [5] for details on those instances constructionprocedure).

The number of nodes of the instances used varies between 50, 75, 100, 125 and 150. For each instance, we considered25%, 50% and 75% of the nodes to be primary. For each set of parameters (number of nodes and proportion of primarynodes), 5 Euclidean instances and 5 random instances were generated. The instances with 50 nodes are the instancesalready used in our previous work [5]. We note, however, that the results reported here regarding the (AArb) modelmay be better than the ones reported earlier. This is due to some improvements on both the model and the machineused.

For every generated instance, we use some of the preprocessing tests proposed by Balakrishnan et al. [3]. Thecomputational times for these tests were negligible for the instances tested. All instances reported in this table have a25% density. For a better understanding of the reported results, instances are divided according to the proportion ofprimary 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 classof instances. In the first column, number of nodes (primary + secondary) and number of edges are given. The numberof primary nodes is also given in percentage in the second column. The third column indicates the type of edge costgeneration method used, either random or Euclidean. The last two columns give similar information to the first twocolumns, after reduction tests have been performed. For example, the first row gives information on instances with 50nodes, 12 primary (which corresponds to 25% of all nodes of that instance) and 38 secondary, and 306 edges generatedby 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 performingthe 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 higherthe percentage of primary nodes is, the higher is the reduction effect. Another interesting fact is that the reduction testsare more effective for Euclidean instances rather than random instances. However, this happens especially when thenumber of nodes increases.

In Table 2, computational results are reported for the 3 compact formulations referred to in this paper. We usedversion 9 of the CPLEX MIP solver for all 3 formulations and we report the CPU time spent, in seconds, until theLP relaxation bound was obtained (and the corresponding gap), and also the CPU time until the integer optimum wasobtained. All tests were performed on a Pentium 4 CPU 3.20 GHz, 512 MB of RAM machine. Instances consideredhave now densities varying between 10%, 25% and 40%. To observe both the effect of the number of nodes of thedensity on the performance of the three models, classes of instances were separated according to those parameters. Thefirst column indicates the number of nodes and the second column indicates the density, respectively, for the group ofinstances considered in each line. For each model ((AArb), (SuperAArb) and (BMM)), we report CPU times to reachthe 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 verytight (for all the cases reported, gaps are equal to zero) and that it improves considerably on the linear programmingbound of the formulation (AArb). However, the improved bounds have the same quality as the bounds given by thelinear programming relaxation of the (BMM) formulation. Additionally, (BMM) takes less CPU time to obtain thelinear programming bounds as well as the integer optimum values. Instances with 150 nodes and density 40% couldnot 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 ofthe instances. The conclusions are similar to the conclusions taken from Table 2. It is interesting to point out thatfor 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

L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611 3609

Table 1Size of the instances after preprocessing

Original Reduced

N(P + S)/M %P (%) N(P + S)/M %P (%)

50(12 + 38)/306 25 Euclidean 47.8(9.8 + 38)/119.6 20Random 47.8(9.8 + 38)/107.8 20

50(25 + 25)/306 50 Euclidean 36.2(11.2 + 25)/84 31Random 37.4(12.4 + 25)/156.8 33

50(37 + 13)/306 75 Euclidean 24(11 + 13)/44 46Random 22.2(9.2 + 13)/60.8 40

75(18 + 57)/693 25 Euclidean 70.8(13.8 + 57)/210 19Random 69.4(12.4 + 57)/507.2 18

75(37 + 38)/693 50 Euclidean 56.2(18.2 + 38)/157.8 32Random 58.8(20.8 + 38)/334.2 35

75(56 + 19)/693 75 Euclidean 34.2(15.2 + 19)/65.6 44Random 31.4(12.4 + 19)/77.6 38

100(25 + 75)/1237 25 Euclidean 82.4(7.4 + 75)/589.2 9Random 92.4(17.4 + 75)/257 19

100(50 + 50)/1237 50 Euclidean 55(5 + 50)/234.6 9Random 74(24 + 50)/176 32

100(75 + 25)/1237 75 Euclidean 28(3 + 25)/72.4 11Random 44.2(19.2 + 25)/75.2 43

125(31 + 94)/1937 25 Euclidean 100.4(6.4 + 94)/810.4 6Random 118(24 + 94)/316.2 20

125(62 + 63)/1937 50 Euclidean 68.4(5.4 + 63)/337.4 8Random 92.2(29.2 + 63)/229.4 32

125(93 + 32)/1937 75 Euclidean 35.2(3.2 + 32)/107.8 9Random 56(24 + 32)/101 42

150(37 + 113)/2793 25 Euclidean 122.2(9.2 + 113)/1170.8 7Random 139(26 + 113)/413.2 19

150(75 + 75)/2793 50 Euclidean 81.8(6.8 + 75)/460 8Random 112.8(37.8 + 75)/292 33

150(112 + 38)/2793 75 Euclidean 40.6(2.6 + 38)/127.4 6Random 68.6(30.6 + 38)/127.2 44

larger than the CPU time provided by the (BMM) model. This can be explained by the fact that, for the instances witha 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 ofthe (BMM) model.

More interesting for the new model is the information given in Table 4 that shows the results in terms of the twodifferent 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 muchworse for random instances.

The CPU times reported for the new model indicate that the CPU times may become critical when solving largerinstances. For such cases, we suggest using, as an alternative, the Lagrangean relaxation scheme proposed in Section4. Therefore, we also tested that procedure in order to evaluate its behaviour for instances that cannot be easily solvedwith the linear programming relaxation.

Table 5 shows some encouraging results for 5 Euclidean instances and 5 random instances with 500 nodes and5000 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, integraloptimality 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.

3610 L. Gouveia, J. Telhada / Computers & Operations Research 35 (2008) 3599–3611

Table 2Results given by the three models according to number of nodes

N D (%) AArb SuperAArb BMM

LP Int. LP Int. LP Int.

Gap (%) CPU (s) CPU (s) Gap (%) CPU (s) CPU (s) Gap (%) CPU (s) CPU (s)

50 10 0.12 0.56 0.76 0.00 0.75 0.84 0.00 0.54 0.6125 0.19 0.59 1.26 0.00 0.76 0.87 0.00 0.52 0.6140 0.42 1.50 4.31 0.00 1.42 1.56 0.00 1.34 1.44

75 10 0.02 4.93 5.57 0.00 6.47 6.79 0.00 5.47 5.7725 0.09 8.30 42.91 0.00 16.04 18.91 0.00 9.40 15.4440 0.33 14.79 204.77 0.00 15.39 16.07 0.00 10.21 10.65

100 10 0.13 16.99 187.51 0.00 45.99 46.61 0.00 19.53 20.0925 0.24 8.17 26.44 0.00 25.44 26.16 0.00 9.27 10.1140 0.35 40.74 230.51 0.00 77.04 79.05 0.00 30.14 31.03

125 10 0.01 50.35 96.17 0.00 76.61 79.72 0.00 27.55 28.9725 0.13 35.98 76.36 0.00 93.10 94.49 0.00 14.84 16.0540 0.32 75.74 3907.71 0.00 274.70 366.66 0.00 68.12 71.28

150 10 0.00 51.99 61.59 0.00 255.36 257.74 0.00 61.10 62.9225 0.17 42.04 187.42 0.00 353.96 475.08 0.00 56.50 59.9240 – – – 0.06 1604.87 3926.62 0.00 994.13 1259.65

Table 3Results given by the three models according to the proportion of primary nodes

%P (%) AArb SuperAArb BMM

LP Int. LP Int. LP Int.

Gap (%) CPU (s) CPU (s) Gap (%) CPU (s) CPU (s) Gap (%) CPU (s) CPU (s)

25 0.38 55.99 945.67 0.00 202.87 245.57 0.00 49.26 52.7350 0.09 12.58 26.94 0.00 39.26 40.45 0.00 12.50 13.1275 0.04 0.33 0.42 0.00 0.35 0.44 0.00 0.32 0.40

Table 4Results given by the three models according to cost generation method

G AArb SuperAArb BMM

LP Int. LP Int. LP Int.

Gap (%) CPU (s) CPU (s) Gap (%) CPU (s) CPU (s) Gap (%) CPU (s) CPU (s)

Euclidean 0.22 15.09 61.98 0.00 12.13 13.74 0.00 14.86 16.00Random 0.14 33.77 622.24 0.00 157.97 187.35 0.00 29.21 31.03

Table 5Some results obtained for instances with 500 nodes

N/M %P (%) Euclidean Random

Iter. CPU (s) Gap (%) Iter. CPU (s) Gap (%)

500/5000 25 3535 6353.8 0.078 8554.2 7596.6 0.04950 1186.6 464 0.000 4491 2634.6 0.08475 2383.4 116.2 0.002 327 59 0.000

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6. Conclusions

Previously, Gouveia and Telhada [5] proposed a formulation for the MWST problem that is not symmetric (thatis, its linear programming relaxation value depends on the choice of the node to be the root of the tree). In thispaper we proposed a new formulation whose linear programming relaxation bound is at least as good as the linearprogramming relaxation bound obtained by using the previous formulation for all possible root selections. The basicidea for obtaining the new formulation is to consider, at the same time, the previous formulation for all possible rootchoices, and guarantee that the feasible solutions of the models associated to every root intersect each other. Our resultsshow that the linear programming relaxation of this so-called “reformulation by intersection” substantially improveson the linear programming relaxation of the original model. The new lower bounds are comparable with the lowerbounds 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 nodesand 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 linearprogramming relaxation of formulations of many other discrete optimization problems, provided that the original givenformulation “suffers” from lack of symmetry.

Acknowledgement

This work was supported by contract grant sponsor: FCT under contract Grant number: Po CTI-ISFL-1-152.

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