Network Flow Models for Designing Diameter-Constrained Minimum-Spanning and Steiner Trees

Luis GouveiaDEIO-CIO, Faculdade de Ciencias da Universidade de Lisboa, Bloco C/2 - Campo Grande, CidadeUniversitaria, 1700 Lisbon, Portugal

Thomas L. MagnantiDepartment of Electrical Engineering and Computer Science and Sloan School of Management, MIT,Cambridge, Massachusetts 02139

We formulate and computationally test several modelsfor the Diameter-Constrained Minimum Spanning andSteiner Tree Problems, which seek a least-cost span-ning or Steiner tree subject to a (diameter) bound im-posed on the number of edges in the tree between anynode pair. A traditional multicommodity flow model witha commodity for every pair of nodes was unable to solvea 20-node and 100-edge spanning tree problem after 1week of computation. In contrast, the new models wereable to optimality solve this problem in less than 1 sec-ond and larger problem instances with up to 100 nodesand 1000 edges. The largest model contains more than250,000 integer variables and more than 125,000 con-straints. The new models simultaneously find a directedtree with a central node or a central edge that serve asa source for the commodities in a multicommodity flowmodel with hop constraints. Our results demonstrate thepower of using single-sourcing combined with other re-formulation techniques: directing the model and usinghop-indexed formulations. Enhancements improve themodels when the diameter bound is odd (these situa-tions are more difficult to solve). The linear programmingrelaxation of the best formulations discussed in this pa-per always give an optimal integer solution for two spe-cial, polynomially solvable cases of the problem. © 2003Wiley Periodicals, Inc.

Key words: spanning trees; Steiner trees; diameter constraints;multicommodity flow models; hop-indexed models

1. INTRODUCTION

We consider diameter-constrained versions of the clas-sical minimal spanning tree (MST) and Steiner tree (ST)problems. Given a prescribed undirected graph G � (V,E)

with node set V and edge set E as well as a cost ce associatedwith each edge e of E, we wish to find an MST or ST witha bound D on the diameter of the tree, which is the maxi-mum number of edges in any of its paths. The diameterconstraints guarantee a specified level of service with re-spect to certain performance measures, for example, theyguarantee a prescribed level of reliability to potential link ornode failures (see, e.g., [21]). When D � 2 or 3, the problemis easy to solve. However, it is NP-hard when D � 4 (see[8]).

In previous studies of this problem, Achuthan et al. [2,3]proposed and tested several exact approaches, solving in-stances with up to 50 nodes and D � 4 and proposed onealternative model, which we briefly summarize in Section3.1. More recently, Abdalla et al. [1] presented some heu-ristics for the diameter-constrained minimum spanning tree(DMST) and examined parallel implementations of theseheuristics.

In this paper, we examine network flow-based formula-tions for the DMST and for the ST version of the problem.For ease of exposition, we cast most of our development forthe MST problem and then later in Section 5 describemodest alterations needed to accommodate the more generalST problem. After formulating a traditional multicommod-ity flow-based integer programming model with a commod-ity for every pair of nodes, we introduce several alternativemodels for the DMST that lead to more effective solutionprocedures. We then report on computational experience,indicating that the modeling improvements can induce verysubstantial reductions in solution time. For example, afterover 1 week of computation, the original formulation wasunable to solve one problem, whereas computations withthe alternative formulations were able to solve the problemin less than 1 second.

Our modeling improvements use four essential ideas:First, following Magnanti and Wong [18], we use an MCF

Correspondence to: L. Gouveia; e-mail: lgouveia@fc.ul.ptReceived March 2000; accepted January 2003Published online 00 Month 2003 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/net.10069© 2003 Wiley Periodicals, Inc.

NETWORKS, Vol. 41(3), 159–173 2003

formulation that has proven to be valuable in modelingmany network design problems. Second, we model theproblem as an MCF problem with a single source, instead ofthe traditional model with a commodity for every pair ofnodes. We simultaneously find a “central node” or a “centraledge” that serves as the source for the commodities. Thismodeling approach permits us to reduce the number ofcommodities by a factor of n, the number of nodes of thegraph. Third, we direct the problem treating the solution asa directed tree from a selected central node or a selectedcentral edge. The idea of directing network design problemshas proven to be a powerful modeling construct in the pastto improve formulations for several network design andconnectivity problems including the ST problem (see[5,9,16,20]) as well as in giving an extended description ofthe convex hull of incidence vectors of spanning trees (see[17,19]). Fourth, following Gouveia [12], we use a hop-indexed formulation to improve the MCF formulation. Hiscomputational results suggest that, in the context of networkdesign problems involving hop constraints, the linear pro-gramming relaxation of an appropriate hop-indexed refor-mulation can improve substantially on the linear program-ming relaxation of an MCF formulation. Our results willdemonstrate the power of using the single-sourcing ap-proach combined with directing the model and using ahop-index formulation for obtaining significant improve-ments in solution times.

The remainder of this paper is organized as follows: InSection 2, we introduce a basic MCF formulation of theDMST problem. In Section 3, we show how to model theDMST as a single-source directed problem, distinguishingsituations in which the diameter restriction D is even andodd. We discuss both traditional network flow and hop-indexed flow models. Our computational results show thatthese models are not as successful for solving the DMSTinstances when D is odd as when D is even. Thus, we alsopresent some valid inequalities for the situation when D isodd. In Section 4, we present a different formulation whenD is odd based upon solving the problem in an expandedgraph containing pseudonodes corresponding to potentialcentral edges. Section 5 briefly discusses minor modifica-tions to the spanning tree formulations for the Steiner case.In Section 6, we report on computation experience ongraphs with up to 80 nodes and 800 edges. These problemsinstances contain up to 250,000 integer variables.

Throughout our discussion, for any optimization modelP, we let PL denote its linear programming relaxation, F(P)denote its set of feasible solutions, and v(P) denote itsoptimal objective value.

2. BASIC FORMULATIONS

To formulate an MCF model of the diameter-constrainedproblem, we associate a commodity with each pair of nodes,defining the set K � {k: k � {p,q}, p,q � V and p � q} ofcommodities with an origin O(k) and a destination node

D(k) for each commodity k. The MCF model uses two setsof variables. Variables xe (e � E) indicate whether the MSTcontains edge e, and directed flow variables y ij

k ({i,j} � E;k � K) specify whether the unique path from the origin ofcommodity k to the destination of commodity k traversesedge {i,j} in the direction i to j. We let n � �V� denote thenumber of nodes in the underlying graph (see Table 1).

This formulation is a multisource multidestination for-mulation for the MST problem, as given in Magnanti andWong [18], augmented with the set of cardinality con-straints �(ij)�E(y ij

k � y jik) � D for all k � K. These inequal-

ities state that the path between any two nodes of V containsno more than D edges. The formulation MCF is quitesimilar to a formulation presented in Balakrishnan andAltinkemer [4] for a more general network design problemwith hop constraints. The MCF model contains a largenumber of variables and constraints and can be difficult tosolve. On a Pentium Pro 450-MHz computer, the CPLEXsolver (version 7.0) was unable to solve a modest-sized20-node and 100-edge instance of the MCF formulationafter 1 week of computation. This model contains about40,000 integer variables and 20,000 constraints.

As noted, the MCF model is based on a multisourcemultidestination network flow model for the MST presentedin Magnanti and Wong [18]. As is well known, we canobtain a single-source multidestination model for the MSTwhose linear programming relaxation is as tight as the linearprogramming relaxation of the multisource multidestinationmodel (see, for instance, [17] and the next section of thispaper). This single-source model involves only n � 1 com-modities, defined from a specific node to all other nodes inthe graph. The “distance” constraints between every pair ofnodes involved in the DMST problem suggest that we mightnot be able to use a similar idea for modeling the DMSTproblem. However, as shown in the next section, specialproperties of trees will permit us to also model the DMSTproblem as a single-source network-flow problem.

TABLE 1. Modeling a diameter-constrained spanning tree.

Model MCF

minimize �e�E

cexe

subject to �e�E

xe�n�1

�j�V

yijk ��

j�V

yjik �� 1 i�O�k�

0 i�O�k�, D�k� for all i�V, k�K�1 i�D�k�

yijk � yji

k �xe for all e�i,j�E,k�K��i,j��E

�yijk �yji

k ��D for all k�K

yijk �0,1 for all i,j�E,k�K

xe�{0,1} for all e�E

160 NETWORKS—2003

3. SINGLE SOURCING AND DIRECTING THEPROBLEM

In this section, we use the following elementary proper-ties of trees (see [14]) to formulate single-source and di-rected versions of the DMST problem. This formulationpermits us to reduce the number of commodities by a factorof n and to obtain much better algorithmic performance:

(i) A tree T has diameter no more than an even integer D ifand only if some node p of T satisfies the property thatthe path from node p to any other node of the treecontains at most D/2 edges.

(ii) A tree T has diameter no more than an odd integer D ifand only if some edge {p,q} of T satisfies the propertythat the path to any other node of the tree from eithernode p or node q contains at most (D � 1)/2 edges.

These properties highlight an intimate relationship be-tween the diameter-constrained spanning tree problem and acenter location problem. Other researchers [7] used themwhen studying a related minimax diameter spanning treeproblem. These properties show that when D is even wecould solve the DMST problem by solving �V� fixed-rootproblems and that when D is odd we could solve theproblem by solving �E� fixed-root problems after shrinkingeach edge to create a pseudonode. We will instead createsingle models that simultaneously select the central nodeand central edge and the diameter-constrained tree.

To ease our notation in describing these models, we willaugment the original network by adding an additional node0 and zero-cost edges {0,j} for all nodes j � V to create anaugmented graph. We will refer to this network as theaugmented network and the network without the node 0 andits incident edges as the original network. We let V0 � V �{0} and E0 � E � {{0,j} for all j � V} denote the node andedge sets in the n � 1 node-augmented network. Then, inthis graph,

(i) When D is even, we wish to select a minimum-costspanning tree containing a single edge {0,j} and ensurethat the unique path from node 0 to every node containsat most (D/2) � 1 edges, and

(ii) When D is odd, we wish to select a minimum-cost setof n � 1 edges so that (a) n edges form a spanning treecontaining exactly two edges {0,i} and {0,j}, (b) theunique path from node 0 to every node contains at most[(D � 1)/2] � 1 edges, and (c) the (n � 1)th edge e� {i,j} from the original network connects the end-points i and j of the two edges {0,i} and {0,j}.

Figures 1 and 2 give examples of the diameter-constrainedspanning tree problems on the augmented networks when D� 4 and D � 5.

In choosing the underlying tree in Figure 1 or 2, we couldchoose n undirected edges or we could replace each edge{i,j} in the network with two directed arcs (i,j) and (j,i),each with the same cost cij as the edge {i,j}. Since we willroot the trees at node 0, we will replace each edge {0,j} with

a single directed arc (0,j). For the directed version of themodels, we let A0 denote the (directed) arc set in theaugmented graph and A denote the (directed) arc set in theoriginal network. In the following sections, we will inves-tigate both the undirected and directed models.

Achuthan et al. [2,3] proposed formulations for theDMST based on a similar augmented network using mod-ified Miller–Tucker–Zemlin subtour elimination con-straints. Although their models are quite compact, resultsgiven in Gouveia [10] using a similar model for the hop-constrained spanning tree problem have shown that thismodeling approach provides weak linear programming re-laxations, for imposing a limit on the maximum number ofarcs in a path. Some results given in Section 6 display thesame computational behavior for the DMST.

3.1. Underlying Core Models

The models presented in this section for the diameter-constrained spanning tree problem contain two core sub-problems when D is even (a rooted MST problem and ahop-constrained path problem) and three core subproblems

FIG. 1. Augumented network for situations with D even (D � 4).

FIG. 2. Augumented network for situations with D odd (D � 5).

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when D is odd including an additional “central edge selec-tion” subproblem. We can model these core models indifferent ways, and by mixing and matching the possiblecore components, we can create several alternate models forthe diameter-constrained spanning tree problem.

Table 2 shows core models for a rooted MST problemdefined on an undirected graph (V0,E0) with n � 1 nodesand a rooted directed MST problem (the minimum arbores-cence problem) defined on a directed graph (V0,A0) with n� 1 nodes. Although the directed graph (V0,A0) could haveany general structure, for our purposes, we obtain it bydirecting a given undirected graph (V,E) as indicated in thelast section. The undirected model uses two sets of vari-ables. Variables xe (e � E) indicate whether the MSTcontains edge e, and directed flow variables y ij

k ((i,j) � A0;k � V; i � k) specify whether the unique path from the rootnode to node k traverses edge {i,j} in the direction i to j. Thedirected model uses two sets of variables as well: thedirected flow variables as in the undirected model andvariables xij ((i,j) � A), indicating whether the spanning treecontains the directed arc (i,j).

The cardinality constraints in these models guaranteethat the tree solution contains n edges (arcs). The inequal-

ities y ijk � y ji

k � xe and y ijk � xij prohibit flowing on any edge

e or arc (i,j) that is not included in the MST. Together withthe flow-conservation constraints, they guarantee that thesolution is connected. Note that the constraints easily implythat �i y ik

k � �i xik � 1for all k � V. Let ST denote thefeasible set of the undirected MST model and DST thefeasible set of the directed MST (directed arborescence)model.

As is well known (see, e.g., [17]), the linear program-ming relaxation of the directed model always has an optimalsolution with integer values for the variables x. The undi-rected model might not have an integer optimal solution. Asnoted by Magnanti and Wolsey [17], it is not difficult toformulate an undirected model whose linear programmingbound is equal to the linear programming bound given bythe directed model. However, the enhanced undirectedmodel contains far more constraints than does the directedmodel and, for our purposes, is impractical from a compu-tational perspective.

To model the hop constraints, we can use either of thetwo models shown in Table 3. In both cases, we assume thatwe are modeling a problem of finding a directed pathcontaining at most H arcs from a given root node (node 0)

TABLE 2. Modeling a rooted spanning tree.

Undirected MST Minimum arborescence model

minimize �e�E0

cexe minimize ��i,j��A0

cijxij

subject to �e�E0

xe � n subject to ��i,j��A0

xij � n

�j�V

yijk ��

j�V0

yjik �� 1 i � 0

0 i � 0, k for all k�V, i�V0

�1 i � k�j�V

yijk ��

j�V0

yjik � � 1 i � 0

0 i�0, k for all k�V, i�V0

�1 i � k

yijk � yji

k �xe for all e � i, j�E0, k�V

yijk � 0 for all e � i, j�E0, k�V

xe � 0,1 for all e � E0

yijk �xij for all�i,j ��A0,k �V

yijk �0 for all�i, j��A0, k�V

xij�{0,1} for all �i, j��A0

TABLE 3. Modeling a hop-constrained path (from node 0 to node k).

Constrained path model Hop-constrained path model

�j�V

z0j1k � 1

�j�V

yijk ��

j�V0

yjik �� 1 i � 0

0 i � 0, k�1 i � k

�j�V

zij2k � z0i

1k for all i�V

�j�V

zijh�1,k � �

j�V

zjihk�0 for all i�V, h�2,,. . . , H � 1

��i,j��A0

yijk � H �

j�V

zjkHk � 1

yijk � {0, 1} for all (i, j) � A0

yijk � �

h�1,. . . ,H

zijhk for all �i, j� � A0

zijhk�0,1 for all �i,j� � A0,h�1, . . . , H

zkkhk�0,1 for all k � V,h�2, . . . , H

162 NETWORKS—2003

to a specific node k. The constrained path model uses onlyflow variables defined on the arcs of the underlying net-work, sending one unit of flow from the root node to nodek, but using at most H arcs. The hop-constrained model is anextended formulation. Besides the flow variables, it alsouses binary variables zij

hk, indicating whether arc (i,j) is thehth arc in the path joining the root node and node k.

Since the 0–1 variables y ijk specify a path between the

root and node k, the inequality �(i,j)�A0y ij

k � H in theconstrained path model restricts the chosen path to containat most H arcs. The hop-constrained path model containsconstraints stating that an arc enters node i in position h ifand only if another arc emanates from this node in positionh � 1 and that one arc enters node k in position H. Note thatthis model contains “loop” variables zkk

hk (h � 2, . . . , H)with zero cost to model situations when the path from theroot node to node k contains fewer than H arcs (i.e., zkk

hk � 1for h � H). The equalities y ij

k � �h�1, . . . H zijhk for all (i,j)

� A0 relate the original variables with the extended vari-ables. Although the original flow variables y ij

k are unneces-sary for obtaining a valid formulation for the core subprob-lem, we will need them for relating the hop-indexedvariables with the flow variables included in the tree mod-els. Let DPk

H denote the set of feasible solutions of theconstrained path model and DHPk

H the set of feasiblesolutions of the hop-constrained model.

The hop-constrained problem contains far more variablesthan does the constrained path problem. However, it is anetwork flow (path) problem on an expanded network andso has the advantage that the extreme points of its linearprogramming relaxation are integer-valued (see [12]),whereas, in general, the linear programming relaxation ofthe constrained path model has fractional extreme points.This result implies that for the linear programming relax-ations, in the space of y variables, the feasible set of thehop-constrained path model is contained in the feasible setof the constrained path model.

The final core subproblem arises only for the situationswhen D is odd and models the selection of the central edgeas well as the selection of two arcs emanating from theauxiliary root node. We wish to choose a single (central)edge {i,j} from the original network as well as two incidentarcs (0,i) and (0,j) from the auxiliary graph. To do so, we letEe be a zero–one variable indicating whether or not wechoose the central edge e � {i,j} to create the edge selection(ES) model in Table 4.

In this model, E(i) denotes the set of original edgesincident to node i. Note that by adding the constraints x0i

� �e�E(i) Ee for all i � V, we obtain �i x0i � 2 �e�E Ee.Then, the equality �e�E Ee � 1 implies that �i x0i � 2 and,consequently, we need not include the constraint �i x0i � 2in the previous model. We also note that the argument justgiven shows that a similar observation holds in the contextof the corresponding linear programming relaxation. As afinal observation, we state the following trivial fact: Thelinear programming relaxation of the ES model gives a

complete description of the convex hull of the integer so-lutions defined by the model.

3.2. Network Flow Models for the DMST Problem

3.2.1. The Situation When D Is EvenAs shown in Table 5, by choosing either direct or undi-

rected versions of the spanning tree problem and the con-strained path or hop variable versions of the hop-con-strained path problem, when D is even, we can create fourdifferent models of the DMST problem.

Our computational results show that the linear program-ming bound given by the undirected multisource/multides-tination MCF model is generally much better than is thelinear programming bound given by the undirected UMCFmodel. However, in some cases, the UMCF model canprovide a tighter linear programming relaxation.

Example. In a complete graph on five nodes numbered 1to 5, suppose that D � 2 and edges {i,j} with j � i � 1 (mod5) have cost 1. Each remaining edge has a cost 10. The MCFmodel has an optimal linear programming value of 7 whilethe UMCF model has an optimal linear programming valueof 8.5. The optimal value to the problem is 22.�

Using the directed DMCF formulation, the CPLEXsolver was able to solve the previously mentioned DMSTinstance with 20 nodes and 100 edges within 200 seconds.For all instances that we tested, the linear programmingbound given by the directed DMCF model is as good(actually better) than is the linear programming model givenby the traditional model MCF. We have been unable toverify whether this relationship holds for all cases. For theprevious five-node example, the DMCF model has a linearprogramming value of 13.0. As in other related models (see[17]), the linear programming relaxation of the directedmodel DMCF is at least as strong as (and generally is muchstronger than) the linear programming relaxation of theundirected model UMCF. Similarly, the directed hop-in-dexed model has a stronger linear programming relaxationthan has the undirected model.

We note that when creating these UMCF and DMCFmodels we can remove one set of flow-balancing constraintsfor each node k � 0 (either from the tree model or from theconstrained path model) since they become redundant. A

TABLE 4. Modeling the ES subproblem.

ES model

�e�E

Ee � 1

�i�V

x0i � 2

x0i � �e�E�i� Ee for all i � Vx0i � 0, 1 for all i � VEe � 0, 1 for all e � E

NETWORKS—2003 163

similar observation applies to the hop-indexed modelsHopUMCF and HopDMCF. Using the equalities yij

k

� �h�1,. . .(D/2)�1 zijhk, it is easy to show that the flow-

conservation constraints of the tree model are aggregationsof the hop-indexed flow-conservation constraints of thehop-constrained path model. Therefore, we can remove theflow-conservation constraints from the tree model. Further-more, the equalities y ij

k � �h�1,. . .(D/2)�1 zijhk relating the two

sets of flow variables permit us to rewrite the linking con-straints (yij

k � xij in the directed model and y ijk � y ji

k � xe inthe undirected model) using the z and x variables and so wecan eliminate the y variables from the hop-indexed models.With these observations, it is easy to show that the directedmodels DMCF and HopDMCF are the same as the modelsdescribed by Gouveia [11,12] for the hop-constrained span-ning tree problem with an additional degree constraint im-posed on the extra node.

Finally, since the extreme points of the linear program-ming relaxation of the hop-constrained path model are in-teger-valued, the feasible points of the linear programmingrelaxation of the constrained path model are contained inthose of the projection of the feasible set of the linearprogramming relaxation of the hop-constrained model.Since adding the directed spanning tree constraints retainsthis relationship, the linear programming relaxation of thedirected hop-indexed model HopDMCF is at least as strongas is the linear programming relaxation of the directedmodel DMCF. Similarly, the linear programming relaxationof the undirected hop-indexed model HopUMCF is at leastas strong as the linear programming relaxation of the undi-rected model UMCF. Formally, we have

Proposition 3.1. v(HopDMCFL) � v(DMCFL) andv(HopUMCFL) � v(UMCFL).

The computational results given in Section 6 indicatethat this dominance is strict for most cases. Using thedirected hop-indexed formulation, the CPLEX solver wasable to solve the previously mentioned 20-node and 100-edge DMST instance in less than 4 seconds.

3.2.2. The Situation When D Is OddWe observed that when D is odd, our models contain

three core subproblems. As in the situation when D is even,we obtain four different models. We might make one otherobservation about these models: If the network contains thecentral edge {j,k}, then it will contain both edges {0,j} and{0,k} and so we can send commodity k directly from node0 to node k and not send any of this commodity on edge{0,j}. Consequently, we can replace the constraint y0

k � x0j

(j � k) with the stronger constraint y0jk � x0j � Ejk (Table 6).

As we observed before, the two directed models satisfythe constraints �i xik � 1 for all k � V. The equalities x0k

� �e�E(k) Ee in the set ES imply that �i�0 xik � �e�E(k) Ee

� 1 for all k � V, that is, every node k � V either receivesan incoming arc from the original network or is an endpointof the central edge. One way to view these models (e.g., byeliminating the variables x0k but retaining the variables y0j

k )is that we are establishing a node (the index 0 plays thisrole) in the middle of the chosen edge (with Ee � 1) with theproperty that the unique path from this node to any othernode contains at most [(D � 1)/2 � 1] edges or arcs. Thisformulation avoids the need to add nodes in the middle ofeach edge of the original network and then treat the addedand original nodes differently since we do not require a pathto added nodes.

As in the D even models, some constraints for the D oddmodels become redundant when we combine the formula-tions for the two core subproblems. As for the D even case,

TABLE 5. Models of the DMST when D is even.

Undirected Directed

Path

Model UMCF Model DMCF

minimize �e�E0

cexe minimize ��i,j��A0

cijxij

subject to (x, y) � ST subject to (x, y) � DST�j�V

x0,j�1 �j�V

x0j�1

y � DP(D/2)�1k for all k � V y � DP(D/2)�1

k for all k � V

Hop

Model HopUMCF Model HopDMCF

minimize �e�E0

cexe minimize ��i,j��A0

cijxij

subject to (x, y) � ST subject to (x, y) � DST�j�V

x0,j�1 �j�V

x0j�1

(zk, y) � DHP(D/2)�1k for all k�V (zk, y) � DHP(D/2)�1

k for all k � V

164 NETWORKS—2003

the hop-indexed models produce a better linear program-ming bound than do the constrained path models. Formally,we have

Proposition 3.2. v(HopDEMCFL) � v(DEMCFL) andv(HopUEMCFL) � v(UEMCFL).

Our computational results show that our models are notas successful for solving DMST instances when D is odd aswhen D is even. In the next subsection, we present a set ofvalid inequalities that permit us to strengthen the linearprogramming relaxation of HopDEMCF.

3.2.3. Valid Inequalities When D Is OddThe inequalities presented in this section are based on an

observation concerning any optimal solution when D is odd,namely, any node linked to the central edge is always linkedto the closest endpoint of the central edge. For every non-central arc (i,j) � A, let L(i,j) denote a set of potentialcentral edges {i,p} � E whose endpoint i is not farther tonode j than is node p, that is, L(i,j) � {{i,p} � E: p � j andcij � cpj}. The previous observation concerning optimalsolutions for situations when D is odd indicates that if anoptimal solution contains an arc (i,j) connected to the centraledge then L(i,j) contains the central edge.

We show next how to use information from the sets L(i,j)to derive valid inequalities for situations when D is odd.These inequalities are based on disaggregated versions ofthe inequalities �j�V z2k

ij � �e�E(i)�{i,k} Ee (for all i,k � Vand i � k) described in the previous subsection. We start byweakening these inequalities into z2k

ij � �e�E(i)�{i,k} Ee forall (i,j) � A and k � V. We can interpret these weakerinequalities as indicating that if arc (i,j) is in position 2 onthe path to node k then the central edge should be adjacentto node i and should differ from the edge {i,k}. However, by

our previous observation concerning optimal solutions forsituations when D is odd, we know that the central edge isrestricted to the set L(i,j)�{i,k}, that is, we can strengthenthese inequalities to z2k

ij � �e�L(i,j)�{i,k} Ee for all (i,j) � Aand k � V.

The new set of constraints permit us to obtain two newmodels for the situations when D is odd. We obtain one ofthe new models, denoted HopDEMCF�, by adding the newconstraints to the model HopDEMCF. We obtain the other,denoted HopDEMCF*, by replacing the O(n2) constraints�j�V z2k

ij � �e�E(i)�{i,k} Ee (for all i,k � V and i � k) inHopDEMCF by the new set of O(n*�A�) constraints z2k

ij

� �e�L(i,j)�{i,k} Ee for all (i,j) � A and k � V. Consequently,the original model HopDEMCF differs from the modelHopDEMCF* only concerning the set of constraints relatingthe hop-indexed variables z2k

ij with the central edge vari-ables Ee. Table 7 depicts the differences between the threemodels.

Since the constraint set of the middle model in Table 7contains the constraints of the other models, we immedi-ately have the following result:

Proposition 3.3. v(HopDEMCF�L) � v(HopDEMCFL)and v(HopDEMCF�L) � v(HopDEMCF*L).

Our computational results show that these inequalitiescan be strict. In general, neither of the linear programmingmodels HopDEMCF and HopDEMCF* dominates theother. Our computational results show that for random costgraphs the new model HopDEMCF* produces a betterbound than does the original model HopDEMCF. The sit-uation is reversed for the instances with Euclidean costs.Using the new model HopDEMCF�, we have been able tosolve problems that we could not solve using the Hop-DEMCF model because its linear programming lower

TABLE 6. Models of the DMST when D is odd.

Undirected Directed

Path

Model UEMCF Model DEMCF

minimize �e�E0

cexe��e�E

ceEe minimize ��i,j��A0

cijxij��e�E

ceEe

subject to (x, E) � ES subject to (x, E) � ES(x, y) � ST (x, y) � DST

y � DP((D�1)/2)�1k for all k � V y � DP((D�1)/2)�1

k for all k � Vy0j

k � x{0,j} � Ejk for all k, j � V; k � j y0jk � x0j � Ejk for all k, j � V; k � j

Hop

Model HopUEMCF Model HopDEMCF

minimize �e�E0

cexe��e�E

ceEe minimize ��i,j��A0

cijxij��e�E

ceEe

subject to (x, E) � ES subject to (x, E) � ES(x, y) � ST (x, y) � DST

(zk, y) � DHP((D�1)/2)�1k for all k � V (zk, y) � DHP((D�1)/2)�1

k for all k � Vy0j

k � x{0,j} � Ejk for all k, j � V; k � j y0jk � x0j � Ejk for all k, j � V; k � j

NETWORKS—2003 165

bound was not tight enough. We note, however, that in somecases, the extra constraints in HopDEMCF� have contrib-uted to huge solution times to obtain the integer solution. Insome of these cases, the third model, the model Hop-DEMCF* which produces a weaker linear programmingbound but is more compact than the strongest model, provedto be a viable alternative for obtaining the optimal integersolution.

To conclude this section, we note that, as shown inGouveia and Magnanti [13], the linear programming relax-ation of the HopDMCF formulation always has an optimalinteger solution when D � 2. When D is odd, the modelHopDEMCF can have fractional extreme points. On theother hand, the linear programming relaxation of the modelsHopDEMCF� and HopDEMCF* always have an optimalinteger solution when D � 3. This observation also impliesthat v(HopDEMCF*L) � v(HopDEMCFL) when D � 3.

4. ENHANCED FORMULATIONS FOR THEDMST WHEN D IS ODD

The formulations described in this section are also basedon our prior observation that in an optimal solution tosituations when D is odd any node linked directly to thecentral edge is always linked to the closest endpoint of thecentral edge. In this section, we use this property to con-struct a different model. We will contract every possiblecentral edge {p,q} into a supernode and replace the two arcsconnecting the nodes p and q to any other node j by theshorter of the arcs (p,j) and (q,j).

This observation permits us to model situations when Dis odd as a special version of the D even problem in anappropriately expanded graph. Let G � (V,E) be the originalundirected graph. We define the expanded graph GE �(VE,EE) as follows:

VE � V � i, j: i, j � E

EE � E � i, j, k: i, j � VE�V, k � V�i, j

and �k, i � E or k, j � E�.

The set VE includes all the nodes in V plus a new set ofnodes, the “supernodes,” corresponding to edges in E. The

set EE includes all the edges in E plus a new set of edgeslinking each supernode to the neighbors of either of theendnodes of the edge corresponding to the supernode. Thenumber of edges in the path from the supernode (say {p,q})to any other node in a tree TE in the expanded graph is atmost (D � 1)/2 if and only if the number of edges in thepath from the central edge {p,q} is at most (D � 1)/2 in aspanning tree T in the original graph.

Based on our experience with the models presented inthe previous section, we shall cast our discussion in terms ofonly directed models. Thus, we replace each edge e � {i,j}� EE of the undirected expanded graph by two directed arcs(i,j) each with the same cost as the edge e and obtain adirected expanded graph GE � (VE,AE). We replace an edgeof the form {{i,j},k} in the undirected graph by only thesingle arc ({i,j},k) in the directed graph. We define the arccost structure in GE as follows: If an arc in GE is of the form(i,j) with i,j � V, then it has the same cost as it has in theoriginal graph G. Otherwise, the arc has the form ({i,j},k)and we set its cost equal to the minimum of cik and cjk. Thedefinition of the new arcs and the way their costs are definedis typical of graph theory algorithms that shrink severalnodes into supernodes (Edmonds’ minimum-cost arbores-cence algorithm is an example). Using the definition of thearc costs in AE, it is easy to transform any feasible tree TE

in GE into a feasible tree T in G with the central edge {p,q}.Furthermore, the definition of the costs in GE guaranteesthat TE is optimal in GE if and only if T is optimal in G.

Following our discussion of Section 3, as illustrated inFigure 3, we will augment the expanded graph by adding anadditional node 0 and zero-cost arcs connecting the rootnode to each supernode. In this augmented graph, we wishto select a tree TE containing a single arc emanating fromnode 0, spanning a single supernode {p,q}, and all theoriginal nodes in V except nodes p and q. The tree mustsatisfy the property that the unique path from node 0 toevery node contains at most (D � 1)/2 � 1 arcs.

The resulting model is similar to the problem discussedin Section 3 for situations when D is even. Indeed, we canwrite the DMCF and HopDMCF models for situations whenD is even in the context of this augmented graph with minormodifications. In this way, we obtain an enhanced directedMST model (Enh-DMCF) and an enhanced directed hop

TABLE 7. The three directed hop-indexed models for D odd.

Model HopDEMCF Model HopDEMCF� Model HopDEMCF*

zii2i � �

e�E�i�

Ee for all i � V zii2i � �

e�E�i�

Ee for all i � V zii2i � �

e�E�i�

Ee for all i � V

�j�V

zij2k � �

e�E�i��i,k

Ee �j�V

zij2k � �

e�E�i��i,k

Ee

for all i, k � V and i � k. for all i, k � V and i � k

zij2k � �

e�L�i,j��i,k

Ee zij2k � �

e�L�i,j��i,k

Ee

for all (i, j) � A and k � V for all (i, j) � A and k � V

166 NETWORKS—2003

indexed flow model (Enh-HopDMCF) for the situationwhen D is odd. For simplicity, we do not formally state thecomplete models or the proof for the following result (see[13]):

Proposition 4.1. v(Enh-DMCFL) � v(DEMCFL) andv(Enh-HopDMCFL) � v(HopDEMCF�L).

As demonstrated by our computational results, the ine-qualities can be strict. Our computational results also showthat the lower bound given by the linear programmingrelaxation of the new model Enh-HopDMCF is, in general,significantly better than is the lower bound given by thelinear programming relaxation of the models described inthe previous section for situations when D is odd. In fact,the lower bounds given by this model for situations with Dodd are close to the optimal integer value and nearly of thesame quality as the lower bounds given by DMCF when Dis even. However, there is an obvious disadvantage to usingthe Enh-HopDMCF formulation. Notice that while the orig-inal graph G has n nodes and m edges, the expanded graphGE contains n � m nodes and O(nm) edges. The number ofvariables and constraints in the new model is substantiallylarger than is the number of variables and constraints in theother models. In fact, our computational results show thateither the linear programming relaxation of the new modelis too large to be solved or the CPU times needed to solveits linear programming relaxation are quite large whencompared with the CPU times of the models described inthe previous section.

Besides being a good choice for solving small-sizedinstances of problems with D odd, we also note that thestudy of this new model together with the study of the D� 3 case (see [13]) suggested the inequalities described inSection 3.2.3. These inequalities have permitted us to solveinstances that are not solved by either the original Hop-DEMCF model (its linear programming lower bound wasnot tight enough) or the enhanced model Enh-HopDMCF(its linear programming model was too large to be solved orrequires too much time to be solved).

5. MINIMUM STEINER TREES WITH DIAMETERCONSTRAINTS

In many applications, the tree need not connect all thenodes of the network. The set of nodes is partitioned intotwo sets, R and S. In telecommunications settings, the ele-ments of the set R usually are terminals that must beconnected to each other and the set S represents switchingnodes. The tree is required to span all nodes in R and mightor might not include some of the nodes in S, the so-calledSteiner nodes. The problem defined in this way is theclassical minimum Steiner tree problem (see [15,17]). Asmentioned in the Introduction, the quality of service re-quirements might suggest a diameter constraint stating thatthe number of edges in the tree path between every pair ofnodes in R does not exceed a given value D. When R � V,we obtain the DMST problem discussed in the previoussections. When �R� � 2, the problem becomes a shortestpath problem between two specified nodes with an addi-tional constraint stating that the path cannot contain morethan D edges. This problem is modeled simply as an un-constrained shortest path problem in an appropriate graphand the solution for this problem provides the underlyingidea for creating the hop-indexed models discussed in Sec-tion 3.

It is easy to modify all the models that we have presentedfor the DMST problem for the Steiner version of the prob-lem: (i) The commodity indices in the formulations rangeonly over the set R instead of the entire node set V and (ii)we replace the constraint stating that the number of edges(arcs) in the spanning tree is equal to �V� � 1 by a constraintstating that the number of edges (arcs) in the tree should begreater or equal to �R� � 1. Because of modification (i),when �R� is small, our models would be small and so moreeasily solvable. Computational results presented in Section6 for the diameter ST problem with �R� � �V�/2 and �R�� �V�/4 confirm that these Steiner instances are easier tosolve than are the corresponding spanning tree versions.

6. COMPUTATIONAL EXPERIENCE

6.1. Problem Instances

To understand what might be achievable with the modelsdiscussed in this paper, we generated several problem in-stances with up to 100 nodes and 1000 edges. We consid-ered two groups of instances: random-cost instances andEuclidean instances. For each value of n (number of nodes)and m (number of edges), we generated different randomand Euclidean instances. We used a uniform distribution inthe interval [1,100] to obtain the cost of each edge includedin each random cost instance. To obtain the Euclideaninstances, we (i) randomly generated the coordinates of nnodes in a 100 � 100 square grid, (ii) selected the cost ofeach candidate edge {i,j} as the integer part of the Euclideandistance between the two nodes i and j, and (iii) defined theedge set E associated with each instance by choosing the m

FIG. 3. Augmented enhanced network for situations with D odd.

NETWORKS—2003 167

least-cost edges of the corresponding complete graph. Thedirected models will contain twice as many arcs as theundirected models. For each instance, we tried several val-ues of the diameter parameter ranging from 4 to 8.

Several of the generated instances are infeasible forsmall values of the parameter D. Thus, for some instances,we omitted the results for D � 4 and 5. The problems areinfeasible because of the way that we selected the edges foreach instance. To address this concern, we conducted a fewpreliminary results with Euclidean instances generated in adifferent manner. First, we select the edges of the minimumstar solution. Then, we select the m � (n � 1) least-costedges of the corresponding complete graph minus the min-imum star edges. Our results showed that these instances aremuch easier to solve than are the ones tested in this paper.We believe that these problems are easier to solve becausethe minimum star solution might give some relevant infor-mation concerning the optimal integer solution for D greaterthan 2. We noticed, for instance, that when D is even theroot of the minimum star is the root of the optimal integersolution of many of the instances tested. We performed alltests on a Pentium II, 450 MHz computer. We used theCPLEX 7.0 package to solve the linear programming mod-els and to obtain the optimal integer values.

6.2. Situation When D Is Even

Tables 8 and 9 summarize our results for the random andEuclidean instances with n � 20 and 30 and D even, that is,D � 4, 6, and 8. The first column specifies the number ofnodes, edges, and required nodes. We chose the set Rarbitrarily. The second column specifies the value of D. Thenext four columns depict the gaps given by the optimallinear programming bound of the models MCF, UMCF,DMCF, and HopDMCF. These columns indicate the value[(OPT � LB)/OPT] � 100 (OPT is the value of the optimalsolution and LB is the value of the lower bound given by theoptimal linear programming solution of the model indicatedat the top of the column). The last column indicates theoptimal value. We restricted the results corresponding toMCF and UMCF to the n � 20 instances and truncated thelower bounds to the first decimal digit. We specify twovalues next to the reported lower bound. The first value isthe CPU time needed to solve the linear programmingrelaxation. To assess the quality of the lower bounds, usingbranch and bound, we tried to obtain the optimal solution oran upper-bound value on the optimum. The second value,when given, is the additional CPU time needed to obtain theoptimal value. CPU times are given in seconds.

TABLE 8. Results for the RANDOM instances with n � 20, 30, and even values of D.

�V�, �E�, �R� D MCF UMCF DMCF HopDMCF Optimum

20,100,10 4 30.0 (25 � 41782) 38.2 (2 � 953) 9.7 (2 � 108) 0.0 (1 � 0) 13820,100,10 6 29.1 (16 �24100) 38.5 (2 � 2098) 2.5 (4 � 60) 0.0 (2 � 0) 11920,100,10 8 29.0 (8 � 9397) 39.3 (2 � 2663) 2.1 (2 � 195) 0.0 (8 � 1) 11520,100,20 4 22.4 (375 � ?) 25.7 (14 � 2530) 13.8 (17 � 1134) 0.0 (4 � 0) 23320,100,20 6 15.3 (68 � ?) 16.8 (6 � 38594) 12.5 (11 � 4400) 1.0 (11 � 35) 17820,100,20 8 4.8 (33 � ?) 5.3 (4 � 12628) 4.0 (6 � 499) 0.0 (10 � 0) 15430,200,15 4 12.4 (29 � 1409) 0.0 (2 � 0) 15930,200,15 6 7.7 (12 � 1371) 0.0 (5 � 0) 10830,200,15 8 0.0 (4 � 0) 0.0 (5 � 0) 9630,200,30 4 11.4 (182 � 7373) 0.0 (6 � 0) 23430,200,30 6 9.5 (69 � 18135) 0.0 (22 � 1) 15730,200,30 8 1.1 (28 � 537) 0.0 (17 � 1) 135

The designation ? indicates that we did not attempt to solve the integer program.

TABLE 9. Results for the EUCLIDEAN instances with n � 20, 30, and even values of D.

�V�, �E�, �R� D MCF UMCF DMCF HopDMCF Optimum

20,100,10 4 12.2 (44 � 38150) 22.4 (3 � 532) 0.8 (2 � 21) 0.0 (0 � 0) 23520,100,10 6 16.2 (26 � 176432) 26.1 (2 � 3778) 0.0 (1 � 0) 0.0 (1 � 0) 21720,100,10 8 22.1 (8 � 86209) 31.4 (2 � 2224) 2.1 (1 � 0) 0.0 (4 � 0) 21720,100,20 4 2.8 (684 � ?) 7.6 (20 � 2790) 1.3 (16 � 186) 0.0 (1 � 0) 36920,100,20 6 2.2 (216 � ?) 6.1 (9 � 2864) 1.0 (15 � 241) 0.2 (6 � 19) 32220,100,20 8 2.5 (97 � ?) 5.4 (6 � 11907) 0.7 (10 � 887) 0.0 (9 � 0) 30830,200,15 4 3.5 (25 � 771) 0.4 (3 � 4) 33830,200,15 6 2.7 (12 � 882) 0.1 (10 � 3) 28930,200,15 8 0.5 (13 � 80) 0.0 (32 � 0) 27430,200,30 4 5.5 (258 � 22179) 1.7 (20 � 1026) 59930,200,30 6 3.2 (298 � 23639) 0.8 (156 � 10899) 48230,200,30 8 3.4 (94 � 21717) 0.8 (319 � 8569) 437

The designation ? indicates that we did not attempt to solve the integer program.

168 NETWORKS—2003

Our results show that the multisource/multidestinationmodel MCF is not able to solve instances with 20 nodes.Single-sourcing allows us to solve many such problems.The reported CPU times indicate that we can solve theseinstances, although with some difficulty, using the undi-rected model UMCF. On the other hand, the directed modelDMCF solves these instances rather easily, but requireshuge CPU times to solve the n � 30 instances. The hop-indexed model is able to quickly solve the n � 20 and 30instances.

We also tested the formulation using the modified Mil-ler–Tucker–Zemlin (MTZ) constraints given in Achuthan etal. [3] for the spanning instances with 20 and 30 nodes. Thisformulation uses a small number of variables and con-straints when compared to the other formulations. Thus, wewere able to obtain the corresponding linear programmingbounds very quickly. On the other hand, these lower boundsare rather weak, and for most of the cases tested, weobtained huge search trees when attempting to obtain theoptimal integer solution. In fact, we could not solve any ofthe Euclidean 30-node instances due to memory-storagelimitations. Moreover, the best lower bound obtained at themoment that we ran out of memory were still far from theoptimal solution (as an example, for the Euclidean instancewith D � 8, the best lower bound was equal to 410 afternearly 40,000 seconds of computation). It is interesting tonote that, for all the cases tested, the MTZ lower bounds areworse than are the trivial lower bounds given by the cost ofthe unconstrained spanning tree. Consider, for instance, theprevious Euclidean example with n � 30 and D � 8. TheMTZ linear programming bound is 340.4 while the MSTbound equals 396.

Table 10 specifies the gaps given by the linear program-ming bound of the hop-indexed model HopDMCF for theinstances with n � 40, 60, 80, and 100. As before, the firstcolumn of this new table indicates the number of nodes,number of edges, and number of required nodes of thecorresponding instance. The next three columns refer to therandom instances and the remaining three columns refer tothe Euclidean instances. Each group of three columns spec-ifies the gaps for D � 4, 6, and 8, respectively. Each entryis similar to the entries in the previous tables, that is, the firstvalue of any cell is the gap multiplied by 100 and the secondvalue is the optimal integer value. The value in parenthesesindicate, respectively, the CPU times for obtaining the lin-ear programming solution and the optimal integer solution.As noted before, and as indicated by the designation INF,the corresponding problem may be infeasible in some cases.

The results show that the HopDMCF model maintainsthe behavior already shown for the smaller instances,namely, that its linear programming bound is quite good.The CPU times required to obtain the optimal linear pro-gramming solution increase with the diameter parametervalue for the Euclidean instances and, in certain cases, arehuge. Notice, however, that the largest model being solved(the HopDMCF model with n � 80, m � 800, �R� � 80 andD � 4) contains about 250,000 integer variables (but far T

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NETWORKS—2003 169

fewer constraints: 130,000). The results also show that theCPU times needed for obtaining the optimal integer solu-tions of the Euclidean instances are, in general, much higherthan the CPU times needed for solving the random in-stances.

6.3. Situation When D Is Odd

For the instances with D odd, we restricted our compu-tational experiments to the models DEMCF, HopDEMCF,HopDEMCF�, HopDEMCF*, and Enh-HopDMCF (the re-sults for D even suggest that the traditional MCF model andthe undirected models are not worth comparing with theprevious four models). Tables 11 and 12 have the sameformat as that of Tables 7 and 8 and contain the results forthe random and Euclidean instances with n � 20 and 30 andD odd, that is, D � 5 and 7. Tables 11 and 12 alsoempirically compare the linear programming relaxations ofthe HopDEMCF, HopDEMCF�, HopDEMCF*, and Enh-HopDMCF models. We indicated the “best” model by des-ignating in bold the CPU times corresponding to the modelthat most quickly produces the optimal integer solution.

The results show that the model HopDEMCF is not assuccessful in solving DMST instances when D is odd as isthe HopDMCF model for solving DMST instances when Dis even. As a consequence, we used valid inequalities forthe D odd case (see Section 3.2.3) and derived the Enh-HopDMCF formulation with all the edges shrunk intonodes. Our results show that, from a theoretical perspective,the linear programming relaxation of this enhanced model is

quite good. In fact, for most of the Euclidean instancestested, this model proved to be the best choice for obtainingthe optimal integer solution. Unfortunately, because theenhanced model contains so many variables and constraints,it may become impractical computationally for larger prob-lem instances. For instance, we encountered computer stor-age limitations when attempting to solve several instanceswith this formulation with n � 40 and m � 400.

The HopDEMCF� model with the addition of n*�A�inequalities implied by the enhanced formulation provides acompromise. Our results for the n � 20 and 30 instancesshow the importance of the HopDEMCF� model whensolving some of the instances with odd values of D. In mostof the cases, the formulation reduced the gap given by theoriginal HopDEMCF model by more than half. For therandom cases, the HopDEMCF� produces zero gaps inseveral cases and proved to be a sound alternative to theenhanced model. Unfortunately, the results are not as goodfor the Euclidean cases. These instances proved to be moredifficult to solve. It is interesting to note that the “alternate”hop-indexed model HopDEMCF* is also worth using forthe random cases. For the Euclidean cases, the linear pro-gramming bounds are quite bad. In fact, in a few situations,it produces a lower bound that is worse than the one ob-tained by the model DEMCF.

The next two tables, Tables 13 and 14, present results forsituations when D is odd for larger instances. We do notreport results from the weak DEMCF model and the hugeenhanced model Enh-HopDMCF (as we have mentionedbefore, we have tried unsuccessfully, to solve some of these

TABLE 11. Results for RANDOM instances when n � 20, 30 and D is odd.

�V�, �E�, �R� D DEMCF HopDEMCF HopDEMCF� HopDEMCF* Enh-HopMCF Optimum

20,100,10 5 6.9 (1 � 145) 0.0 (0 � 0) 0.0 (0 � 0) 0.0 (0 � 0) 0.0 (11 � 1) 12020,100,10 7 9.7 (1 � 33) 5.1 (1 � 41) 0.0 (1 � 0) 2.7 (1 � 33) 0.0 (70 � 1) 11620,100,20 5 22.3 (8 � 4190) 8.5 (2 � 132) 2.4 (4 � 84) 4.1 (2 � 57) 0.0 (49 � 1) 20520,100,20 7 12.2 (5 � 1241) 6.7 (4.4 � 272) 1.5 (8 � 47) 4.3 (3 � 215) 1.5 (442 � 310) 16530,200,15 5 19.4 (8 � ?) 9.2 (2 � 95) 3.7 (5 � 115) 4.7 (3 � 104) 0.0 (157 � 4) 13230,200,15 7 3.7 (6 � ?) 2.5 (3 � 3) 0.0 (6 � 0) 0.0 (3 � 1) 0.0 (880 � 4) 9830,200,30 5 20.4 (43 � ?) 6.6 (7 � 1211) 0.0 (9 � 1) 2.8 (5 � 245) 0.0 (462 � 7) 19530,200,30 7 7.8 (25 � ?) 4.4 (16 � 957) 0.0 (29 � 107) 0.5 (19 � 207) 0.0 (4228 � 7) 144

The designation ? indicates that we did not attempt to solve the integer program.

TABLE 12. Results for EUCLIDEAN instances when n � 20, 30 and D is odd.

�V�, �E�, �R� D DEMCF HopDEMCF HopDEMCF� HopDEMCF* Enh-HopMCF Optimum

20,100,10 5 8.8 (1 � 71) 6.8 (0 � 40) 0.0 (2 � 0) 4.9 (11 � 1) 0.0 (10 � 0) 22520,100,10 7 6.3 (1 � 92) 5.5 (1 � 60) 1.7 (9 � 314) 7.3 (2 � 417) 0.0 (142 � 1) 21720,100,20 5 8.7 (12 � 3855) 4.6 (3 � 179) 2.2 (7 � 238) 4.9 (2 � 287) 0.0 (41 � 1) 34720,100,20 7 6.1 (6 � 1219) 3.7 (6 � 958) 1.1 (14 � 414) 3.2 (4 � 541) 0.0 (395 � 1) 31630,200,15 5 12.2 (19 � ?) 6.7 (4 � 1004) 2.4 (24 � 806) 11.8 (4 � 1156) 0.0 (95 � 3) 30930,200,15 7 7.6 (10 � ?) 5.1 (25 � 964) 2.1 (78 � 1238) 9.4 (9 � 1914) 0.0 (6976 � 3) 27830,200,30 5 11.2 (104 � ?) 5.8 (22 � 8257) 4.2 (61 � 17947) 10.2 (7 � 31496) 0.1 (195 � 6) 53430,200,30 7 9.8 (44 � ?) 5.7 (47 � 102013) 3.8 (140 � 62216) 6.0 (32 � 77941) 1.2 (31168 � 24129) 463

The specification ? indicates that we did not attempt to solve the integer program.

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instances). For the Euclidean instances, we do also not showthe results from the model HopDEMCF* as the results ofTable 12 indicate that this model behaves rather poorly forthis class of instances. The results demonstrate the relevanceof the models developed in Section 3.2.3. The HopDEMCF�and HopDEMCF* models (this one in the context of therandom instances) helped to solve instances that we werenot able to solve with the original hop-indexed model Hop-DEMCF. Unfortunately, with these improvements, our bestmodels for the situations when D is odd are not of the samequality level as are our models for the situations when D iseven. As before, we indicated the “best” model by desig-nating in bold the CPU times corresponding to the modelthat most quickly produces the optimal integer solution. Wenote that the CPU times needed for obtaining the optimal

integer solution depend strongly on the behavior of theinteger programming package, namely, in how it obtains agood upper bound. In several cases, the linear programmingbound given by the model HopDEMCF� is at least as goodas is the best lower bound obtained by the original modelHopDEMCF after 1 day of computation of the branch-and-bound code. However, in a few of these cases, the strength-ened HopDEMCF� requires a reasonable amount of time toobtain the optimal integer solution because it finds a goodupper bound rather late in the optimization process. Oneexample of this is the instance with n � �R� � 60, D � 7 and300 edges. After 10,000 seconds of computation of thebranch-and-bound method using the HopDEMCF� model,the best lower bound was already within one unit of theoptimal integer solution. Unfortunately, the computations

TABLE 13. Results for RANDOM instances when n � 40 and D is odd.

�V�, �E�, �R� D HopDEMCF HopDEMCF� HopDEMCF* Optimum

40,400,20 5 7.5 (14 � 737) 2.5 (22 � 890) 6.5 (10 � 372) 12840,400,20 7 9.4 (39 � 1381) 2.9 (99 � 749) 6.0 (29 � 233) 10140,400,40 5 9.1 (56 � 13529) 5.6 (104 � 23645) 7.5 (40 � 8642) 25340,400,40 7 1.0 (114 � 209) 0.0 (174 � 4) 0.0 (69 � 4) 17160,600,15 5 16.1 (7 � 667) 6.0 (13 � 640) 9.4 (9 � 477) 8860,600,15 7 9.0 (34 � 769) 1.6 (139 � 183) 7.7 (25 � 715) 6160,600,30 5 10.9 (33 � 12240) 4.1 (62 � 3318) 7.0 (30 � 6425) 16060,600,30 7 8.4 (150 � 5313) 3.5 (395 � 7093) 6.4 (90 � 5897) 10960,600,60 5 14.8 (142 �t) 10.0 (280 � t) 13.5 (96 �t) 257 u60,600,60 7 7.2 (686 � 61113) 2.9 (1795 � t) 6.0 (551 � 85466) 15080,800,10 5 16.9 (5 � 188) 9.7 (6 � 234) 14.5 (5 � 184) 4680,800,10 7 14.7 (15 � 294) 0.0 (89 � 2) 5.8 (18 � 1011) 3480,800,20 5 9.4 (17 � 2252) 4.8 (33 � 2405) 8.6 (19 � 2455) 11180,800,20 7 8.0 (164 � 4408) 3.8 (253 � 3049) 8.5 (114 � 8836) 7880,800,40 5 12.3 (74 � 50355) 7.8 (124 � 103318) 12.2 (58 � 19673) 18680,800,40 7 3.6 (418 � 25887) 0.0 (1306 � 10) 2.8 (362 � 120191) 114100,1000,12 5 5.1 (7 � 483) 0.0 (7 � 3) 0.0 (8 � 9) 45100,1000,12 7 4.2 (30 � 1038) 0.0 (59 � 4) 2.5 (38 � 773) 40100,1000,25 5 7.9 (40 � 7179) 1.2 (67 � 152) 7.4 (23 � 1675) 97100,1000,25 7 6.3 (145 � 7074) 1.4 (577 � 2217) 3.3 (110 � 2530) 71

The designation t indicates that we did not obtain the optimal solution after 2 days of CPU time. The designation u indicates that the value shown onthe right-hand column is an upper bound since none of the three models was able to solve the corresponding problem.

TABLE 14. Results for EUCLIDEAN instances when n � 40 and D is odd.

�V�, �E�, �R� D HopDEMCF HopDEMCF� Optimum

40,400,20 5 6.3 (32 � 3921) 3.6 (71 � 7815) 36240,400,20 7 4.4 (136 � 7962) 2.1 (1052 � 27565) 31740,400,40 5 4.1 (185 � 52493) 2.4 (641 � 81911) 61240,400,40 7 4.8 (1021 � t) 3.3 (3770 � t) 527 u60,600,15 5 6.4 (36 � 3386) 3.1 (76 � 5720) 30760,600,15 7 6.6 (193 � 50002) 4.3 (1004 � 46240) 28060,600,30 5 7.7 (142 � 98422) 5.0 (697 � t) 58560,600,30 7 5.9 (992 � t) 4.3 (5443 � t) 481 u80,800,10 5 13.0 (10 � 1666) 7.7 (38 � 10625) 24180,800,10 7 7.0 (100 � 3832) 3.0 (566 � 29579) 21480,800,20 5 11.1 (63 � 27029) 7.8 (161 � t) 47380,800,20 7 11.1 (958 � t) 8.7 (3569 � t) 401

The designation t indicates that we did not obtain the optimal solution after 2 days of CPU time. The designation u indicates that the value shown onthe right-hand column is an upper bound since none of the three models was able to solve the corresponding problem.

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yielded an upper bound corresponding to the optimal valuerather late in the optimization method yielding about 90,000seconds of CPU time.

6.4. Summary and Conclusions

Our computational results led to the following conclu-sions:

1. Using the models presented in the paper, we were ableto solve Steiner instances containing up to 100 nodesand 1000 edges (with �R� � (1/2) �V� and D � 8) andspanning tree instances with up to 60 nodes and 600edges and with D � 8.

2. Euclidean problems appear to be more difficult than arerandom instances.

3. For situations when D is even, the linear programmingrelaxation of the model HopDMCF produces gaps ofless than 1% for almost all problem instances.

4. Situations when D is odd appear to be more difficultthan do those when D is even.

4a. The linear programming gaps for the models range froma few percent to as much as 17%, with gaps of 5–8%being typical.

4b. For these instances, the models HopDEMCF,HopDEMCF�, and HopDEMCF* compete for gen-erating solutions most quickly.

7. CONCLUSIONS

We have introduced single-source models for the diam-eter constrained minimum spanning tree and Steiner treeproblems. A traditional model introduces a commodity andimposes a hop constraint for each pair of required nodes. Bysimultaneously selecting a central node or a central edgeand using it as source for the commodities in an MCFmodel, our models contain only n commodities.

Our best model, which introduces a directed version ofthe model with hop constraints, was able to solve a small-sized instance in less then 1 second that the traditionalmodel has not been able to solve after 1 week of computa-tion. We presented computational results for spanning treeand ST instances with up to 100 nodes and 1000 edges andwere able to solve a model with slightly more than 250,000integer variables and 250,000 constraints. To the best of ourknowledge, our results provide the first computational ex-perience in solving the diameter-constrained ST problem.

By using an auxiliary augmented network, we developedformulations for situations when D is even and D is odd. Asshown in Gouveia and Magnanti [13], it is fairly easy toobtain equivalent formulations (in terms of correspondinglinear programming relaxations) involving only variablesassociated with the original network. They also showed thatthe linear programming relaxation of the best models dis-cussed in this paper always produce an integer optimalsolution for two easily solved cases of the DMST (thosewith D � 2 and D � 3).

Our progress in improving the algorithmic performance

of solution methods for this class of problems rests uponseveral modeling ideas: exploiting the underlying graphstructure, using MCF models for network-design problems,directing network-design models, introducing extended (inthis case hop-indexed) formulations, and using polyhedralmethods (valid inequalities). As in other application con-texts reported in the literature, our experience illustrates thevalue of embracing a multifaceted modeling approach forsolving (mixed) integer programming models.

Acknowledgments

The authors are grateful to the referees for their sugges-tions leading to a substantially improved presentation of theresults of this paper. The authors are particularly grateful toone of the referees whose comments have led us to cast ourmodels using the auxiliary network.

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