Multistars and Directed Flow Formulations

Luis GouveiaDEIO–CIO, Faculdade de Ciencias da Universidade de Lisboa, Bloco C/2 - Campo Grande, CIDADEUNIVERSITARIA, 1749-016 Lisbon, Portugal

Leslie HallDepartment of Mathematical Sciences, Johns Hopkins University, Baltimore, Maryland 21218

The Capacitated Minimum Spanning Tree Problemseeks a least-cost spanning tree subject to a boundimposed on the number of nodes in each subtree pend-ing from a given root node. Araque et al. (TechnicalReport SOR-90-12, Princeton University, 1990) intro-duced several classes of facet-defining inequalities forthe undirected version of the problem, most of whichhave straightforward analogs to the directed version andare also facet-defining in that case (see Zhang, Master’sthesis, 1993). The multistar constraints are one suchclass. Gouveia [Telecommun Syst 1 (1993), 51–56]showed that a directed flow formulation gives a polyno-mial representation of the class of directed multistarconstraints. This equivalence shows how to obtain apolynomial-time separation algorithm for this class ofinequalities. In this paper, we show that the previousequivalence result implies that we can also separate inpolynomial time the exponential-sized class of undi-rected multistar constraints. We also show that “using adirected model” plays a key role in obtaining a polyno-mial-time separation algorithm for this class of inequal-ities, that is, using a directed flow model seems to becrucial for obtaining a polynomial-time separation algo-rithm for the class of undirected multistar constraints.© 2002 Wiley Periodicals, Inc.

Keywords: capacitated trees; integer programming; polyhedralprojection; separation algorithms

1. INTRODUCTION

A basic problem in network design for telecommunica-tions applications is the problem of designing a centralizedcomputer network. We are given a central processor and aset of remote terminals that must be connected to the centralprocessor. The problem is to design a network connecting

the terminals with the central processor that has the follow-ing properties: (1) The network is a tree; (2) the number ofnodes in any subtree off the central processor node is notgreater than a given integer K � 1 (note that when K � 1,the problem either is trivial or infeasible); and (3) the totaldesign cost of the network is minimized, where the cost ofthe network is equal to the sum of the costs of each of thelinks included in the network. Like most network designproblems, this problem [known as the Capacitated Mini-mum Spanning Tree (CMST) Problem] has long beenknown to be intractable (Papadimitriou [17]).

To derive an integer programming formulation for thisproblem, we assume that the central processor and terminalsare nodes of a complete graph G � (V, E). A “natural”formulation for this problem contains exactly one variablefor each edge of the underlying network: The binary deci-sion variable ue, for e � E, indicates whether or not toinclude edge e in the network design. In this case, however,this so-called natural formulation appears to require anexponentially large set of constraints to model the connec-tivity requirements associated with the spanning tree designand the capacity constraints on the subtrees off the root (seeSection 3). An alternative approach is to derive a so-calledextended formulation which uses additional variables (seePulleyblank [18] for a formal definition of what is meant bya natural versus an extended formulation of a combinatorialoptimization problem). These additional variables may beconsidered superfluous in the sense that they are not neces-sary for expressing a valid formulation of the problem. Onthe other hand, the additional structure imparted by includ-ing them may considerably reduce the number of constraintsneeded for a valid model, that is, by using a few extravariables, we might obtain a compact formulation with apolynomial number of constraints and variables, as opposedto the exponential number of constraints required originally.In this study, we focus on a particular type of extendedformulations, so-called network-flow formulations, whichare a common tool in modeling network design problems

Received January 2000; accepted July 2002Correspondence to: L. Gouveia e-mail: lgouveia@fc.ul.ptPublished online 00 Month 0000 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/net.10050© 2002 Wiley Periodicals, Inc.

NETWORKS, Vol. 40(4), 188–201 2002

(see, e.g., Magnanti and Wong [15]). The additional flowvariables allow the connectivity and capacity constraints tobe modeled very compactly. Perhaps the most basic exam-ple of this phenomenon is the uncapacitated minimum span-ning tree problem (see Magnanti and Wolsey [14]).

Another way of enhancing the natural formulation is byconsidering the underlying graph to be directed rather thanundirected, thus doubling the number of design variables:The design variable for each undirected edge e � {i, j} isreplaced by two variables corresponding to directed arcs (i,j) and ( j, i), both with cost coefficients equal to that of theoriginal variable. The additional structure imposed by di-recting the arcs often allows one to obtain quite easily astronger model. Of course, this same enhancement can beapplied to the extended flow formulation. Again, the unca-pacitated minimum spanning tree problem gives a goodexample of a wide variety of insights for “directing” anoriginal undirected model (see Magnanti and Wolsey [14]).Goemans and Myung [7] presented and discussed interest-ing relationships between the linear programming relax-ations of several natural and extended formulations, for theSteiner tree problem, both in the undirected and in thedirected space. Interesting results on directing a model alsoappeared in Magnanti and Raghavan [13]: Nontrivial ideasfor directing network design problems with higher connec-tivity requirements and for directing the Steiner forest werediscussed and presented.

This paper discusses several models for the CMST prob-lem: “natural” directed and undirected models and the di-rected and undirected models extended with (single-com-modity) flow variables. In a certain way, we can say that aprevious result from Gouveia [8], which showed that adirected flow formulation gives a polynomial representationof the exponential-sized class of directed multistar con-straints, was, perhaps, a trigger for the research in thispaper. This equivalence result shows how to obtain a poly-nomial-time separation algorithm for this class of inequal-ities (the result and its consequences are summarized inSection 2). In this paper, we show that the previous equiv-alence result implies that we can also separate the exponen-tial-sized class of undirected multistar constraints. This isshown by equating the linear programming relaxation oftwo natural formulations: a directed formulation and anundirected formulation (these results are presented in Sec-tion 3). These results motivate, in turn, the following ques-tion: Does “directing the model” play a key role in obtainingthe polynomial-time separation algorithm for the class ofundirected multistar inequalities? In Section 4, we givestrong arguments indicating that this is the case, that is,using a directed flow model seems to be crucial for obtain-ing a polynomial-time separation algorithm for the class ofundirected multistar constraints. We demonstrate that thedirected flow formulation has a nontrivially stronger LPrelaxation than its undirected counterpart: A small set ofvalid inequalities for the former implies an exponentiallylarge class of valid inequalities for the latter. We also

demonstrate that these inequalities define facets of the un-directed flow polytope. Taken together, these results offercompelling evidence that directed formulations are nontrivi-ally stronger than are their undirected counterparts. Theseresults help us to understand and appreciate the value inconsidering alternative formulations for a problem.

For the problem considered here, the flow formulationsdescribed in the paper are not computationally competitive,and computational work was not our motivation for study-ing these models. The best approaches that we know forsolving this problem to optimality, in practice, are basedeither on branch-and-cut for the formulation in undirectedvariables (Hall [12]) or on a different extended formulation(Gouveia and Martins [10]). Although several other papersdiscussed in detail the theoretical relationships betweendirected and undirected formulations (see, for instance, theprevious mentioned papers [7, 13, 14]), we believe that ourpaper is the first one which discusses the same issue in thecontext of a capacitated network design problem. The threepreviously mentioned papers presented results comparingundirected with directed multicommodity flow formulationsfor several uncapacitated network design problems. Theseresults have one common feature: Although the undirectedformulation contains more constraints than does the directedformulation, the number of constraints in the undirectedformulation is still polynomial in the number of nodes andedges of the input graph. The results of the present papershow that whenever capacity constraints are considered inthe context of a directed multicommodity flow model (thereare several applications motivating the use of such con-straints in such models—in fact, one can formulate theCMST as a multicommodity flow model), it is much morecomplicated to obtain an equivalent undirected multicom-modity model. In fact, our results also suggest that such anundirected formulation would contain an exponential-sizedset of constraints.

2. MOTIVATION

We start by introducing our notation: Let node 0 be thecentral processor, or “root” node, and V � {1, . . . , n}denote the set of terminal nodes. Since in our model alltraffic is generated at the root and flows toward the termi-nals, we shall assume that no traffic flows into the root node.We let V0 � V � {0} denote the set of all nodes in theunderlying network. Finally, we let E denote the set of allpossible edges in the network and ce denote the cost ofincluding edge e in the network.

As we previously pointed out, we can formulate theCMST problem using directed edges (arcs), that is, we candirect all the edges of the tree paths from the root node toevery other node in the tree. In the directed formulation,every edge {i, j} of the graph is replaced by the two arcs (i,j) and ( j, i). For all edges, we set the arc costs cij � cji

equal to the cost c(i, j). We let xij be the binary decisionvariable indicating whether or not to include the directed arc

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(i, j) in the tree and let yij denote the flow of traffic on arc(i, j). Because of our assumption that no traffic flows backinto the root node, we can assume, from now on, that yi0

� 0 and xi0 � 0 for all i � V or, equivalently, that thesevariables simply do not exist. We consider the problem offinding a directed spanning tree, that is, a spanning treewhose paths from the root to every other node are directedaway from the root. This feature can be ensured by requiringthe indegree of every nonroot node to be exactly 1 and byassociating directed flow with its corresponding directeddesign arc. The Directed Flow (DF) formulation describedbelow was first suggested by Gavish [5]:

Formulation DF (Gavish [5])

min �i�V0

�j�V

cijxij

s.t. �i�V0

xij � 1 j � V, (1)

�i�V0

yij � �i�V

yji � 1 j � V, (2)

y0j � Kx0j j � V, (3a)

yij � �K � 1� xij i, j � V, (3b)

yij � xij i � V0, j � V, (4)

xij � �0, 1� i � V0, j � V. (5)

The flow variables yij and arc design variables xij with i� j are ignored in the model. The flow conservation con-straints (2) combined with the so-called forcing constraints(3a, b) ensure connectivity of the network. These two sets ofconstraints combined with the indegree constraints (1) guar-antee that the solution is a directed spanning tree. The“upper bounding” constraints (3a) also ensure that the ca-pacity constraints are enforced: At most, K units of flow areallowed on any link of the solution emanating from the rootnode, which implies also that each subtree off the rootcontains at most K nodes. We obtain the linear program-ming relaxation of DF by replacing (5) with the lower andupper bounding constraints:

0 � xij � 1 i � V0, j � V. (5�)

By projecting out the variables yij from the polyhedrondefined by (1)–(4) and (5�), we can obtain a new formula-tion using the variables xij alone and such that its linearprogramming bound is equal to the linear programmingbound of the directed formulation DF.

To introduce the new formulation, we describe next the

directed version of a set of constraints called multistarinequalities. Araque et al. [2] introduced several classes offacet-defining inequalities for the undirected version of theproblem, most of which have straightforward analogs to thedirected version and are also facet-defining in that case (seeZhang [19]). The multistar constraints are one such classand their directed version is as follows:

�i, j�S

Kxij � �i�S, j�V�S

� xij � xji� � �S��K � 1�,

S � V, �S� � 1. (6)

As we shall see in Section 3 (when interpreting thecorresponding undirected version), these inequalities can beseen as generalized degree constraints on the node set S.

We shall call (1), (5), and (6) the Directed Multistarformulation and its linear programming relaxation is ob-tained by replacing (5) with (5�). The following result is dueto Gouveia [8] and shows an intimate relationship betweenthe directed multistar constraints and the directed DF for-mulation.

Proposition 1. Consider a solution x satisfying (1) and(5�). Then, x satisfies the directed multistar constraints (6) ifand only if there is a flow y such that the pair (x, y) satisfiesconstraints (2), (3a, b), and (4).

The proof of the above result, given in Gouveia [8], isbased on ensuring that a feasible flow exists in a given graphand highlights the fact that each multistar constraint is, infact, a cut condition on the corresponding node set S. Whatis more, the previous result implies that the separationproblem for the exponential-sized class of the directed mul-tistar constraints can be solved in polynomial time.

Corollary 1. The directed flow formulation DF provides apolynomial-time algorithm for solving the following sepa-ration problem: Given a solution x satisfying (1) and (5�),determine whether x satisfies all of the inequalities (6), andif not, find one such inequality.

The result follows from the fact that to check (in poly-nomial time) whether an arbitrary vector x satisfying (1) and(5�) satisfies the multistar constraints (6) we simply writedown the associated flow problem induced by the givendirected solution x as defined in the proof of Proposition 1(see Gouveia [8]) and then determine (in polynomial time)whether or not this flow problem has a feasible solution. Ifit is not feasible, then the algorithm will indicate a cutwhose capacity is too small (see, for instance, Ahuja et al.[1] on obtaining a cut-set from a flow solution); this cut, inturn, indicates a set S for which the constraint (6) is vio-lated.

Note that in the statement of Corollary 1 the solution x isnot even required to satisfy (1) and (5�). As inequalities (1)

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and (5�) can be separated in polynomial time, the separationproblem for inequalities (6) can be solved in polynomialtime with respect to any solution x.

For the results of Section 3, it is relevant to considerdirected models with the following inequalities:

xij � xji � 1 i, j � V, i � j, (7)

which are clearly valid for directed formulations. In whatfollows, directed models with constraints (7) will be de-noted, respectively, by the Enhanced Directed Flow (EDF)model and the Enhanced Directed Multistar model.

The results of this section suggest the following ques-tion: Can we also separate the exponential-sized class ofundirected multistar constraints? In the next section, weshall give an affirmative answer to this question. In fact, weshall show that Proposition 1 with constraints (7) added tothe two directed models plays an important role in obtaininga polynomial-time algorithm for separating the undirectedmultistar constraints.

The affirmative answer to the previous question moti-vates the following new question: Does “directing themodel” play a key role in obtaining the polynomial-timeseparation algorithm for the class of undirected multistarinequalities? In Section 4, we shall give strong argumentsindicating that this is the case, that is, using a directed flowmodel seems to be crucial for obtaining a polynomial-timeseparation algorithm for the class of undirected multistarconstraints.

3. SEPARATING THE UNDIRECTEDMULTISTARS

In this section, we shall present an undirected formula-tion whose linear programming relaxation is equivalent tothe linear programming relaxation of the enhanced directedmultistar formulation. We will also show that this equiva-lence result implies that the EDF formulation provides apolynomial-time separation algorithm for the undirectedmultistar constraints. Let ue be the binary decision variableassociated with edge e. Occasionally, we will find it usefulto write u(i,j) for the edge e � {i, j}. We consider the setunordered, and, thus, we equivalently write u(i,j) or u( j,i). Con-sider, then, the following Undirected Multistar formulation:

Formulation Undirected Multistar

min �e�E

ceue

s.t. �e�E

ue � n, (8)

�e�E�S�

ue � �S� � 1 S � V0, 0 � S, �S� � 2 (9)

�e�E�S�

Kue � �e�E�S,V�S�

ue � �S��K � 1� S � V, �S� � 1,

(10)

ue � �0, 1� e � E. (11)

It can be easily shown that this formulation is a valid mixedinteger programming formulation for the CMST problem.We note that inequalities (9) are not even needed for ob-taining a valid formulation for the CMST. However, theyneed to be considered in the undirected model if we want toshow that the linear programming relaxation of this model isequivalent to the linear programming relaxation of the en-hanced directed multistar model.

Constraints (8) state that the solution has n � �V0� � 1edges. In the “tree inequalities” (9), E(S) denotes the set ofedges, whose endpoints are in S. These inequalities are sonamed because they comprise a subset of the inequalitiesneeded to describe the uncapacitated spanning tree polytope(Edmonds [4]). Whenever 2 � �S� � n � 1, the treeinequalities define facets of the capacitated tree polyhedron(see Hall [11]).

Constraints (10) are the undirected multistar inequalities(see Araque et al. [2]) which were mentioned in Section 2.To understand why these constraints are valid for the undi-rected capacitated spanning tree polyhedron, we first notethat, for the case �S� � 1, the constraint says that a terminalnode can be connected to at most K � 1 other terminalnodes, since at most K nodes can be clustered in any subtreeoff of the root. Next, consider the case �S� � 2. Let S � { p,q} and consider the edge e � { p, q}. To see how large theleft-hand side of (10) can be, we consider two cases: First,suppose that ue � 0. Then, it is possible to connect up to K� 1 nodes to p and to q, resulting in a left-hand-side valueof at least 2(K � 1). Next, suppose that ue � 1. Then,since p and q are connected, the pair can be directly con-nected to at most K � 2 other nodes. Again, this results ina left-hand-side value of at least K K � 2 � 2(K � 1)� �S�(K � 1). This argument is easily generalized to largersets S (see Araque et al. [2]).

We obtain the linear programming relaxation of theundirected multistar formulation by replacing (11) with thelower and upper bounding constraints:

0 � ue � 1 e � E. (11�)

Let us now consider the relative strengths of the linearprogramming relaxations of the two formulations: the en-hanced directed multistar formulation and the undirectedmultistar formulation. More formally, we wish to comparethe sets of feasible solutions of the directed and undirectedformulation by equating the variables in the obvious way:

u�i, j� � xij � xji i, j � V, i � j, and

u�0, j� � x0j j � V. (12)

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We start by providing the following lemma:

Lemma 1. Consider directed and undirected solutions xand u related by (12). Then, x satisfies (1), (5�), and (7) ifand only if u satisfies (8), (9), and (11�).

Proof. First note that x satisfies (5�) and (7) if and onlyif u satisfies (11�). Consider a directed solution satisfying(1), (5�), and (7). By summing the directed degree con-straints (1) for j � 1, . . . , n and using (12), we obtain theequality (8). Consider, now, any subset of nodes S thatincludes the root node. By summing equalities (1) over thenodes of S other than node 0, we have

�S� � 1 � �j�V0

�i�S��0�

xji � �j�S

�i�S��0�

xji � �j�S

�i�S��0�

xji

� �j�S

�i�S��0�

xji � �e�E�S�

ue.

In the final equality, we used (12) and the fact that xi0 � 0for all i � V. This shows that any directed solution satis-fying (1), (5�), and (7) maps to an undirected solutionsatisfying (8), (9), and (11�).

To show the other direction, consider a solution u that isfeasible to the undirected model defined by constraints (8),(9), and (11�). We need to show that there exists a way ofallocating u(i, j) as xij xji, for all pairs i, j, so that theresulting directed solution x satisfies (1), (5�), and (7),which is equivalent to showing that the following bipartitemaximum flow problem has a maximum flow of n units (seeFig. 1):

(i) There is a source node s, a sink node t, and a node foreach edge e � E with an arc from the source node swhose capacity is se � ue.

(ii) There is a demand node for each node i � V with anarc to the sink node t with capacity 1;

(iii) For each edge e � {i, j} � E, there are arcs from the“left-hand” node corresponding to e to the “right-hand” nodes i and j.

(iv) A capacity infinity is associated with any edge betweenthe left-hand and the right-hand sets.

We interpret the flow from {i, j} to i as representing thevalue of the variable xji, while the flow from {i, j} to jrepresents the value of the variable xij. In this case, then,constraint (1) for node i is enforced by the capacity con-straint of value 1 on the arc from i to t, since the flow intonode i is given by ¥j�0, . . . ,n xji. Similarly, (7) is enforcedby the capacity constraint on the arc from s to the “left-hand” node {i, j}, since the sum of the flows from {i, j} toi and to j is bounded above by the flow from s to {i, j},which is bounded above by u(i, j) � 1. Each arc has a lowercapacity of zero. Clearly, this maximum (s–t)-flow problemhas a flow of value n if and only if the desired allocationexists, since, if the desired allocation exists, it induces a maxflow of ¥e�E xe � n units of flow through the max-flownetwork and conversely. Now, we need to argue that themax-flow value of this network is in fact n or, equivalently,that the minimum capacity of any cut separating s from t inthe network is equal to at least n. Let S be the set of verticesin the max-flow network representing edges (i.e., the “left-hand” set of nodes), and let T represent the “right-hand”nodes, that is, nodes corresponding to nodes of V. Consider

FIG. 1. The max-flow network and a cut with finite capacity.

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an arbitrary cut [N, N�] in the network with finite capacity,where s � N and t � N�. Since arcs from S to T haveinfinite capacity, there can be no arcs from S � N to T �N�.

Thus, the capacity of the cut can be expressed as¥e�S�N� ue �N � T�. Moreover, by our assumption thatthe cut is finite, for all {i, j} � S � N, i, j 0, we musthave i � N � T and j � N � T, and for {0, j} � S �N, we have j � N � T. These facts imply that

�e�S�N

ue � �e�E��T�N���0��

ue.

However, the tree constraint (9) for N � T � {0} tells usthat

�e�S�N

ue � �e�E��T�N���0��

ue

� ��T � N� � �0�� � 1 � �T � N�.

Thus,

�e�S�N�

ue � �N � T� � �e�S�N

ue � �e�S�N�

ue � �e�E

ue � n,

as we wished to show. �

The previous lemma has several relevant consequences.First, the bipartite maximum flow problem used in the proofof the lemma shows how to transform an undirected solu-tion satisfying (8), (9), and (11�) into a directed solutionsatisfying (1), (5�), and (7). In a similar manner, we see thatthe separation problem for the tree inequalities (9) is solvedin polynomial time.

The fact that the n � 1 directed degree constraints (1)imply the sum constraints (8) and the exponential-sized setof constraints (9) in the undirected space is known (see, forinstance, Hall [12] and Magnanti and Wolsey [14]). Thelemma shows that (8) and (9) are the only constraints thatcan be generated by the degree constraints (1). This equiv-alence helps in relating the LP relaxations of some undi-rected and directed models in the design variables that havealready been proposed for the capacitated spanning treeproblem. We can make the following general statement:Suppose that we have a set of inequalities UI(u) defined onthe undirected u variables and a corresponding set of ine-qualities DI( x) defined on the directed x variables, obtainedby replacing each occurrence of u(i, j) by the substitutiongiven in (12) for all unordered pairs i, j. Clearly, if adirected solution x is feasible for DI( x), the associatedsolution u is feasible for UI(u) and conversely. Moreover,an undirected solution u is feasible for UI(u), (8), (9), and(11�) if and only if a directed solution x satisfying (12) isfeasible for DI( x), (1), (5�), and (7).

One example of such a set UI(u) is given by the so-

called packing inequalities (or “generalized” subtour elim-ination constraints). For simplicity, we omit the descriptionof these inequalities from this paper. We remark, however,that the argument just given shows that the linear program-ming relaxation of the undirected model described in Gav-ish [6] is equivalent to the linear programming relaxation ofdirected formulations described in several papers in theliterature (see, for instance, Gouveia [8, 9] and Hall [12]).

A second example of a set UI(u) is given by the multi-star constraints (10). Lemma 1 and the arguments just givenprove the following:

Proposition 2. Consider directed and undirected solu-tions x and u related by (12). Then, x is feasible for thelinear programming relaxation of the enhanced directedmultistar formulation if and only if u is feasible for thelinear programming relaxation of the undirected multistarformulation.

We noted that Proposition 1 still holds if inequalities (7)are included in both models: the directed flow model DFand the directed multistar formulation. Then, Proposition 2implies the following result:

Corollary 2. An undirected solution u is feasible for thelinear programming relaxation of the undirected multistarformulation if and only if there exists a directed solution inthe design variables x and a flow y such that x and u arerelated by (12) and (x, y) is feasible for the linear program-ming relaxation of the enhanced directed flow formulationEDF.

The previous result implies that the separation problemfor the exponential-sized class of the undirected multistarconstraints is also solvable in polynomial time. Consider asolution u satisfying (8), (9), and (11�). The proof of Lemma1 shows how to transform u into a directed solution xsatisfying (1), (5�), and (7). Then, by Corollary 1, we cancheck in polynomial time whether x satisfies the directedmultistar constraints (6). If x satisfies all of these con-straints, then the solution u also satisfies the undirectedconstraints (10). On the other hand, if x violates someconstraints in (6), then the relation (12) gives the corre-sponding undirected multistar constraints (10) violated byu. Thus, we have the following corollary:

Corollary 3. The directed flow formulation DF provides apolynomial-time algorithm for solving the following sepa-ration problem: Given a solution u satisfying (8), (9), and(11�), determine whether u satisfies all of the inequalities(10), and if not, find one such inequality.

Remarks similar to the ones made after presenting Cor-ollary 1 can also be made in this case. The solution u is notrequired to satisfy (8), (9), and (11�). If u does not satisfy(9), then it is not possible to send n units of flow from s to

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t in the bipartite graph of Lemma 1. A minimum capacitycut permits us to identify indegree constraints (1) whichmust be violated by a directed solution x satisfying (12).The violated inequalities can be added to the model and weobtain a directed solution satisfying (1), (5�), and (7). Sim-pler observations apply to the case where u does not satisfy(8) or (11�).

The result stating that the directed flow formulation givesa polynomial-time algorithm for separating the undirectedmultistar constraints permits us to cast the question given atthe end of Section 2 in the following way: Can we find anundirected flow formulation whose linear programming re-laxation is equivalent to the linear programming relaxationof the undirected multistar formulation? More importantly,can we find a formulation in undirected variables that,besides the tree inequalities (9), is a compact (polynomial-sized) formulation that answers the previous question affir-matively? Note that this is equivalent to finding an undi-rected compact formulation whose linear programmingrelaxation is equivalent to the linear programming relax-ation of a weaker undirected multistar formulation withoutthe tree inequalities. In the next section, we examine undi-rected flow formulations and we give strong argumentsshowing that such a formulation may not exist.

4. UNDIRECTED FLOW MODELS

4.1. A Basic Model

We start by presenting a straightforward undirected flowmodel that is similar to one discussed and presented in Hall[11]. The model uses undirected ue variables as in the modeldescribed in the previous section and directed flow variablesyij as in the directed flow model:

Formulation UF (Hall [11])

min �e�E

ceue

s.t. �e�E

ue � n, (8)

�i�V0

yij � �i�V

yji � 1 j � V, (2)

y0j � Ku�0, j� j � V, (13a)

yij � yji � �K � 1�u�i, j� i, j � V such that �i, j� � E,

(13b)

yij � 0 i, j � V, (14)

ue � �0, 1� e � E. (11)

The flow conservation constraints (2) combined with theforcing constraints (13a, b) ensure connectivity of the net-work. Constraints (13b) are bidirectional upper boundingconstraints and their validity follows from the fact that, inany tree solution containing a given edge {i, j}, at most oneof the variables yij and yji will be positive. Constraints (2)and (13a, b) together with (8) ensure that the solution is aspanning tree, since it requires the solution to contain n� �V0� � 1 edges. Constraints (13a) ensure that the capac-ity constraints are enforced.

As before, we obtain the linear programming relaxationof UF by replacing (11) with the lower and upper boundingconstraints:

0 � ue � 1 e � E. (11�)

Constraints (13b) are reminiscent of an idea first given inBalakrishnan et al. [3] for tightening the forcing constraintsin a multicommodity flow model. An undirected flow modelwith a weaker linear programming relaxation can be ob-tained by using instead the weaker unidirectional upperbounding constraints:

yij � �K � 1�u�i, j� i, j � V.

We note that we have not included lower boundingconstraints [as we did in the directed model with constraints(4)] in the undirected formulation because the “natural”analog for the undirected formulation,

yij � u�i, j� i, j � V, (15)

is not even valid, since for a given edge {i, j}, we may haveyij � 0, while u{i, j} � 1. However, if u{i, j} � 0, then atleast one of the arcs (i, j) or ( j, i) must carry flow. Thus,we can write the valid bidirectional lower bounding in-equality

yij � yji � u�i, j� i, j � V such that �i, j� � E. (16)

It is not hard to see, however, that these constraints areentirely redundant in the linear programming relaxation ofUF in the sense that if a solution (u, y) violates some ofthese constraints then it is possible to find another solution(u, y�) satisfying all constraints (16). To see this, considera solution (u, y) that is feasible for the linear programmingrelaxation of UF but which does violate constraints (16) forsome edges {i, j}. Such a solution can be modified tosatisfy the previously violated lower bounding constraints(16) by simply sending additional flow around length-twocycles, that is, for each edge {i, j} corresponding to aviolated constraint, we increase the values of the variablesyij and yji in equal amounts, until the lower boundingconstraint (16) is satisfied as an equality constraint. Such atransformation will not make the solution infeasible for the

194 NETWORKS—2002

following reasons: First, flow conservation will be main-tained. Second, the bidirectional upper bound constraints(13b) are not in danger of being violated, since, after addi-tional flow has been added, we have yij yji � u{i, j} �(K � 1)u{i, j}.

There seems to be no straightforward way of capturing inthe undirected version the lower bounding informationgiven by (4). Indeed, we suspect that there is no “easy” wayto capture such information, that is, there is no polynomial-size set of constraints strictly on the undirected variablesthat captures it. We will have more to say about this later on,when we exhibit a rather unintuitive set of inequalities thatappear to represent the lower bound constraints in the un-directed model, in the sense that they are obtained byprojecting the directed lower bound constraints.

What is more is that the next result shows that thedirected upper bounding constraints (3b) also capture moreinformation than do the undirected bidirectional upperbounding constraints (13b):

Proposition 3.

(i) Consider a feasible solution (x, y) to the linear program-ming relaxation of the directed formulation EDF withoutthe lower bounding constraints (4) and the undirected (u,y) solution obtained from the directed solution by usingthe linear transformation (12). Then, (u, y) is feasible forthe linear programming of the undirected formulation UFaugmented with the tree inequalities (9).

(ii) There may exist solutions (u, y) that are feasible for thelinear programming of the undirected formulation UFaugmented with the tree inequalities (9) with the propertythat there is no feasible solution (x, y) for the linearprogramming relaxation of EDF without the lower bound-ing constraints (4) such that u and x are related by (12).

Proof.

(i) By combining the directed upper bounding constraints(3b) for the two given pairs (i, j) and ( j, i) and by using

(12), we obtain the bidirectional undirected upperbounding constraint (13b) for the corresponding edge{i, j}, and, thus, the undirected solution satisfies all thebidirectional undirected upper bounding constraints(13b). Obviously, (3a) together with (12) implies theupper bounding constraint (13a). The remainder of theproof follows from observations given in the proof ofLemma 1.

(ii) Consider the following example on seven nodes with K� 3 (see Fig. 2) and the following undirected solution:u{0,1} � u{0,2} � u{0,3} � u{1,4} � u{2,5} � u{3,6}

� u{4,5} � u{4,6} � u{5,6} � 2

3with flows y01 � y02

� y03 � 2 and y14 � y25 � y36 � 1. All othervariables are zero. It is easy to verify that this examplesatisfies the tree inequalities (9) and all the inequalitiesin the UF model. On the other hand, there is no way toconstruct a directed solution ( x, y) with the propertythat x satisfies the transformation (12) and the solution( x, y) satisfies the directed formulation. To see this,note that there is a unique way to resolve the x variablesin order to satisfy the degree constraints (1) associatedwith nodes 1, 2, and 3, namely, x41 � x52 � x63 � 1

3,

forcing x14 � x25 � x36 � 1

3.

This implies that there is not enough capacity to carrythree units of flow from the set of nodes {1, 2, 3} to the set{4, 5, 6}, namely, at most two units of flow can cross the cutseparating the two node sets. Hence, there is no y-vectorsuch that ( x, y) is feasible for the directed model. �

This strict dominance implies, in effect, that couplingflow variables with directed design variables somehow pro-vides more combinatorial structure than does coupling flowvariables with undirected design variables. In the next sec-tion, we shall use subtler arguments to exhibit additionalclasses of inequalities for the undirected model, derivedfrom the directed model, which “cutoff” the solution in theseven-node example.

4.2. Generalized Upper and Lower Bounding Constraints

As we have shown in Section 3, interesting inequalitiesfor undirected models can be generated from directed mod-els. In this section, we shall focus on flow-based inequali-ties. To motivate our arguments, consider the bidirectionalcoupling constraints (13b) introduced for the undirectedmodel in Section 4.1. A simple argument was given to provetheir validity for the undirected model. However, with re-spect to other inequalities, it might be difficult to find sucha direct proof or, more importantly, to even generate suchnew inequalities. An alternative way of generating inequal-ities (13b) would have been to follow the proof of Propo-sition 3 and then somehow invoke the directed upper bound-ing constraints (3b). In this section, we use similar but moresubtle arguments to generalize the undirected upper bound-ing constraints and show how to obtain lower boundingconstraints for undirected models.

FIG. 2. Value of edges shown in the solution is equal to 2

3. Values of flow

variables are depicted next to the edges and the flow is from left to right.

NETWORKS—2002 195

Let S � V. First, we observe that by adding the degreeinequalities (1) for all j � S we obtain

�i, j�S

xij � �i�V�S, j�S

xij � �j�S

x0j � �S�,

which can be rearranged as

�i�V�S, j�S

xij � �S� � �i, j�S

xij � �j�S

x0j. (17)

Suppose, now, that we sum up the directed upper boundingconstraints (3b) over i � V�S and j � S. Then, we obtain

�i�V�S, j�S

yij � �K � 1� �i�V�S, j�S

xij

� �K � 1���S� � �i, j�S

xij � �j�S

x0j�.

The final equality in the expression follows from (17). Byrearranging terms and using the variable substitution (12),we have the following inequality on the undirected vari-ables:

�i�V�S, j�S

yij � �K � 1� �e�E��0��S�

ue � �K � 1��S� S � V.

(18)

The inequalities above are denoted as Generalized UpperBounding (GUB) constraints. If, for example, we let S� {4, 5, 6} in the example on seven nodes given in Section4.1, we see that this new valid inequality is violated.

In a similar manner, we can derive a set of undirectedGeneralized Lower Bounding (GLB) constraints. For S �V, if we sum up the directed lower bounding constraints (4)over i � V�S and j � S (and note that 0 � V�S), we obtain

�i�V�S, j�S

yij � �i�V�S, j�S

xij � �S� � �i, j�S

xij � �j�S

x0j,

where the final equality follows from (17). Again, by rear-ranging terms and using the variable substitution (12), thefollowing undirected inequality is obtained:

�i�V�S, j�S

yij � �e�E��0��S�

ue � �S� S � V. (19)

We have proved the following proposition:

Proposition 4. Consider a feasible solution (x, y) to thelinear programming relaxation of EDF and the undirected(u, y) solution obtained from the directed solution by usingthe linear transformation (12). Then, the GUB constraints

(18) and the GLB constraints (19) are satisfied by theundirected solution (u, y).

We shall next show that we can generate the undirectedmultistar constraints (10) from the GUB constraints (18),the GLB constraints (19), the upper bounding constraints(13a) for edges incident to the root node, and the “sum”constraints (8):

Proposition 5. Consider an undirected solution u satisfy-ing (8) and (11�). Then, u satisfies the multistar constraints(10) if there is a flow y such that the pair (u, y) satisfies theflow conservation constraints (2), the GUB constraints (18),the GLB constraints (19), and the upper bounding con-straints (13a).

Proof. Consider a set S � V. First, by adding (13a)over all j � S and then adding the result to the GUBconstraints (18) for the same set S, we obtain the followinginequality:

�i�V0�S, j�S

yij � �e�E��0��S�

ue � K �e�E�S�

ue � �K � 1��S�. (20)

By summing up the flow conservation constraints (2) overall j � S and eliminating like terms, we obtain

�i�V0�S, j�S

yij � �S� � �j�S,i�V�S

yji.

Applying (20) to the set S and (19) to the set V�S andcombining, we obtain

�K � 1��S� � �e�E��0��S�

ue � K �e�E�S�

ue � �S�

� �V�S� � �e�E��0��V�S�

ue,

which can be rearranged to give

�K � 1��S� � �V� � K �e�E�S�

ue

� � �e�E��0��V�S�

ue � �e�E��0��S�

ue�.

Observe also that the “sum” constraints (8) can be rewrittenas

n � � �e�E��0��V�S�

ue � �e�E��0��S�

ue�

� �e�E�S,V�S�

ue for all S � V.

By combining these last two expressions and noting that �V�

196 NETWORKS—2002

� n, we obtain the undirected multistar constraint (10) forthe same set S. �

Notice that in the previous result the bidirectional con-straints (13b) were not even needed to derive the multistarconstraints. Observe also that the same result also holds ifthe tree inequalities (9) are included in both models.

By using a circular chain of dominance/equivalence re-sults, we can show that the previous result, with the treeinequalities included, holds as an equivalence result:

Proposition 6. Consider an undirected solution u satisfy-ing (8), (11�), and the tree inequalities (9). Then, u satisfiesthe multistar constraints (10) if and only if there is a flow ysuch that the pair (u, y) satisfies the flow conservationconstraints (2), the GUB constraints (18), the GLB con-straints (19), and the upper bounding constraints (13a).

Proof. The observation just given after Proposition 5guarantees that the linear program corresponding to theundirected flow model defined by (8), (9), (2), (11�), (13a),(18), and (19) dominates the linear programming relaxationof the undirected multistar formulation. Moreover, Corol-lary 2 equates this linear programming relaxation with thelinear programming relaxation of the directed flow modelEDF. Finally, Propositions 3 and 4 show that (12), (13a),(18), and (19) are implied by inequalities in this directedmodel. �

Figure 3 depicts the propositions used in the circularargument of the proof of Proposition 6.

Proposition 6 implies the following “near-equivalence”between the directed and undirected flow models, namely,

they are equivalent with respect to their design variables,but not necessarily their flow variables:

Corollary 4. Consider directed and undirected designvariables x and u, respectively, related by (12). Then, thereexists a flow vector y such that (u, y) satisfies the undirectedflow model defined by (8), (9), (2), (11�), and (13a) plus theGUB and GLB inequalities (18) and (19), if and only if thereexists a flow vector y� such that (x, y�) is feasible for thelinear programming relaxation of the EDF formulation.

Observe that it is not necessarily the case that y � y�.Moreover, notice that to establish this equivalence, theundirected upper bounding constraints (13b) are not neededin the undirected model.

Corollary 4 has an interesting implication, namely, con-straints (13a) and the GLB and GUB constraints are the onlynonredundant constraints that can be obtained by adding uparbitrary constraints from the sets (3a, b) and (4) and trans-forming to the undirected space. Moreover, the result high-lights the elegance and simplicity of instead focusing on thedirected formulation.

Finally, we have the following interesting result, whoseproof is provided in the Appendix. We define the undirectedflow polytope as the convex hull of feasible (undirected)trees plus flows that send one unit of flow to each nonrootnode along the tree, that is, it is the convex hull of theinteger solutions to the original flow formulation UF.

Proposition 7. The GUB constraints (18) and the GLBconstraints (19) define facets of the undirected polytope,provided that S and V�S are not empty, �V� � 2 and K � n.

This result implies that the directed upper and lowerbounding constraints contain information that, when pro-jected on the undirected flow space, necessarily requires anexponential number of inequalities to represent. As a con-sequence, it is highly unlikely that we can find a formulationin undirected variables that besides the tree inequalities (9)is a compact (polynomial-sized) formulation and whoselinear programming relaxation is equivalent to the linearprogramming relaxation of the undirected multistar formu-lation.

5. CONCLUSIONS

In this paper, we have shown that a directed single-commodity flow model for the CMST contains informationthat, when projected on the undirected flow space, neces-sarily requires an exponential number of inequalities torepresent. We have shown that the directed model providesa polynomial-time separation algorithm for the exponential-sized class of the undirected multistar inequalities and alsothat it is highly unlikely that such an algorithm could havebeen induced by an undirected flow formulation.

As we have pointed out in the Introduction, for the

FIG. 3. Circular chain of main results.

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problem considered here, the flow formulations described inthe paper are not computationally competitive, and compu-tational work was not our motivation for studying thesemodels. We wish to emphasize that we have made this studyto better understand the structure of directed versus undi-rected models for a capacitated network design problem.We believe that our work represents some important firststeps in understanding the theoretical relationships amongthese formulations.

Acknowledgments

The authors are grateful to Douglas Shier and to thereferees for their suggestions leading to a substantiallyimproved presentation of the results of this paper.

APPENDIX: PROOF THAT THE GLB AND GUBINEQUALITIES ARE FACET-DEFINING

We begin by restating Proposition 7:

Proposition 7. The GUB constraints (18) and the GLBconstraints (19) define facets of the undirected polytope,provided that S and V�S are not empty, �V� � 2 and K � n.

Proof. Let A be the set of integer solutions (u, y)representing feasible trees plus flows that send one unit offlow to each nonroot node along the tree, that is, A is the setof integer solutions to the integer program UF. We wish toestablish that these inequalities define facets of the polyhe-dron P � conv( A) defined as the convex hull of feasibletrees with flows in the (u, y) space. We will prove this factunder some mild assumptions on the sizes of the sets S andV�S. The proof given here will require �V�S� � 2 and �S� �2; however, as stated in the proposition, the result actuallyholds as long as S and V�S are nonempty and �V� � 3.These small-cardinality cases need slightly modified argu-ments that are not hard but are tedious and will thus beomitted.

Before we begin, we remind the reader of our notationalconventions: when referring to an undirected edge betweennode i and j, we use the notation {i, j}, whereas a directedarc (or flow) from i to j is denoted (i, j). For disjoint sets T,U � V, the notation E(T, U) can refer either to the set ofundirected edges between T and U or to the set of arcsdirected from T to U, depending on the context. Similarly,the set E(U) refers either to the set of undirected edges orto the set of directed arcs with both endpoints in U, depend-ing on the context.

The technique that we will use in our proof is thewell-known dual approach for establishing such results, andwe describe it briefly here. The polyhedron P has severalinherent equality constraints; in particular, every point of Psatisfies the flow conservation constraint at each node, plusthe constraint ¥eue � n.

Suppose that we have an inequality au by � d that wewish to show is facet-defining for P � conv( A). First, wemust show that au by � d is valid (which, in our case,has already been established in Section 4). We must alsodemonstrate that at least one point (u, y) � A satisfies au by � d, since, otherwise, the inequality is just a linearcombination of the inherent equality constraints of themodel. This, too, is obviously true for the GUB constraints(18) and the GLB constraints (19).

What remains to be shown is that the intersection of thehyperplane defined by au by � d with the polyhedron Phas dimension at least (and, hence, exactly) one less than thedimension of P. The dual approach for proving this fact isbased on the following idea (see, e.g., Nemhauser andWolsey [16]):

Consider any valid inequality for P, �u y � , withthe property that every point (u, y) � A satisfying au by� d also satisfies �u y � . Then, (a, b, d) definesa facet of P if and only if, for every such (�, , ), the vector(�, ) is a linear combination of (a, b) and the inherentequalities of P.

We begin by proving that the GLB inequalities arefacet-defining. Recall the GLB inequality:

�e�E��0��S�

ue � �i�V�S, j�S

yij � �S� S � V. (19)

Let S0 � S � {0}. In the case of the GLB constraints, toprove that every such inequality �u y � is in theequivalence class, it suffices to show that it is possible tochoose multipliers for the inherent equality constraints andadd these multiples to (�, , ) so that

(a) For all e � E(S0), the �e values are equal; we will callthis value �.

(b) For all e � E(S0), �e � 0.(c) For all arcs (i, j) � E(V�S, S), ij � 0.(d) For all arcs (i, j) � E(V�S, S), ij � �.

If (a)–(d) hold for (�, ), then, clearly, (�, ) is a multipleof (a, b). Thus, we shall have shown that (a, b, d) isfacet-defining. It is interesting to note that we have made noclaims about whether there might be additional inherentequalities for our polyhedron, and, indeed, we do not needto; however, the course of our proof actually shows that thisis not the case and, thus, simultaneously establishes thedimension of P.

The way that we prove (a)–(d) is by invoking inter-change arguments: We consider two very similar integersolutions of P that both satisfy au by � d; infer that �u y � for both solutions; subtract one equality from theother; and obtain a result that gives us a relationship amonga few of the coefficients of (�, ). As we proceed, we willmodify (�, ) by adding multiples of the equality con-

198 NETWORKS—2002

straints as necessary to “zero out” certain sets of the coef-ficients.

Part (a): Show that the �e values are equal for all e� E(S0). First, fix i, j � S. Consider the following two treesolutions (u1, y1) and (u2, y2) that satisfy au by � d:

(i) The star solution consisting of edges {0, k} for k� 1, . . . n, (i.e., u{0,k}

1 � y0k1 � 1, for k � 1, . . . , n);

(ii) The previous star solution with edge {i, j} replacingedge {0, j} (i.e., u{0,k}

2 � 1 for k � 1, . . . , n; k j,u{i, j}

2 � 1, y0i2 � 2, yij

2 � 1, and y0k2 � 1 for all k

i, j).

By observing that both satisfy �u y � and subtract-ing one equation from the other, we have

��0, j�u�0, j�1 � 0jy0j

1 � 0iy0i1 � ��i, j�u�i, j�

2 � 0iy0i2 � ijyij

2 � 0

or, equivalently,

��0, j� � ��i, j� � 0i � ij � 0j. (21)

Now, consider another pair of solutions (u3, y3) and (u4,y4), based on fixing a third node h � V:

(i) The star solution with edge { j, h} replacing edge {0,h} (i.e., u{0,k}

3 � 1 for k � 1, . . . , n; k h, u{ j,h}3

� 1, y0j3 � 2, yjh

3 � 1, and y0k3 � 1 for all k j, h);

(ii) The previous solution with edge {i, j} replacing edge{0, j} (i.e., u{0,k}

4 � 1 for k � 1, . . . , n; k h, j,u{ j,h}

4 � u{i, j}4 � 1, y0i

4 � 3, yij4 � 2, yjh

4 � 1, andy0k

4 � 1 for all k j, h, i).

Subtracting the (�, ) equations from one another forthese two solutions yields

��0, j� � ��i, j� � 2�0i � ij � 0j�. (22)

Subtracting (21) from (22) gives (0i ij � 0j) � 0and, as a consequence [by using (21) and (22)], we obtain�{0, j} � �{i, j}. It is straightforward to conclude, from thefact that i and j were chosen arbitrarily from S, that �e is thesame for all edges of E(S0). We shall refer to this commonvalue as �.

Part (b): Show that �e � 0 for all e � E(S0) [afteradjusting (�, ) with multiples of the equality constraints].First, by choosing i, j � V�S (and h � V), we can use anargument identical to that in part (a) to show that, for alledges e � E(V0�S), the values of �e are equal. Now, byadding an appropriate multiple of ¥eue � n to the vector(�, , ), we can make this common value zero. [Notice that� defined in part (a) will change, but that all coefficientsoriginally equal to � will remain so.]

Next, we show that edges of E(S, V�S) also have �e

� 0. Let i � V�S, j � S and consider the two solutions (u1,y1) and (u2, y2):

(i) The star solution;(ii) The star solution with edge { j, i} replacing edge {0, i}.

Both solutions satisfy au by � d and, therefore, alsosatisfy �u y � . Subtracting �u2 y2 � from�u1 y1 � yields

��0,i� � �� j,i� � 0j � ji � 0i. (23)

Next, fix a third node h � V�S, and consider the followingpair of solutions (u3, y3) and (u4, y4):

(i) The star solution with edge {0, h} removed and edge{i, h} added;

(ii) The previous solution with edge { j, i} replacing edge{0, i}.

Subtracting �u4 y4 � from �u3 y3 � yields

��0,i� � �� j,i� � 2�0j � ji � 0i�. (24)

Combining (23) and (24) gives �{0,i} � �{ j,i}. Recall thatwe have already shown that �{0,i} � 0, since i � V�S.Moreover, i and j were chosen arbitrarily from V�S and S,respectively, and, thus, �{ j,i} � 0 for all edges { j, i}� E(S, V�S).

Part (c): Show that jk � 0 whenever ( j, k) � E(V�S,S), (after adjusting (�, ) with multiples of the equalityconstraints). We begin by fixing i, j � V� S, and k � V0,and we consider two integer solutions (u1, y1) and (u2, y2)of the following form:

(i) The star solution except that edges {0, i} and {0, j} arenot in the tree and, instead, edges {k, i} and {k, j} are(if k � 0, we just take the star solution);

(ii) The previous solution with edge {k, j} removed andedge {i, j} added instead.

By subtracting �u2 y2 � from �u1 y1 � andrecalling that �e � 0 for all e � E(S0), we find that

kj � ki � ij � 0 for all i, j � V�S and k � V0. (25)

Next, we fix i�, j� � V�S, and we note that i� and j� willremain fixed throughout the rest of the part (c) argument.First, by adding a multiple of the flow conservation con-straint on node j� to (�, ), we make i�j� � 0. (Observethat this adjustment does not affect any component of �.) Bytaking i � i� and j � j� in (25), we obtain kj� � ki� forall k � V0�{i�, j�}. By adding multiples of the flowconservation constraints on each node k � V0�{i�, j�}, wecan now simultaneously make

kj� � ki� � 0 for all k � V0��i�, j��. (26)

Holding i� and j� fixed, we will now show that, in fact, all

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other relevant -coefficients are also zero: We will showthat kh � 0 for all arcs (k, h) � E(V�S), (k, h) � E(S0,V�S), and (k, h) � E(S0), where, in each case, k, h i�,j�. Also, we will show that i�k � j�k � 0 for all k � V�S,k i�, j�. Consider any pair of nodes h � V�S, k � V0.By (26), hj� � ki� � 0. Using this fact, and taking i � hand j � j� in (25), we obtain 0 � kj� � kh hj�

� kh.Since h and k were chosen arbitrarily from V�S and V0,

respectively, we see that kh � 0 for all arcs (k, h)� E(V�S) and all arcs (k, h) � E(S0, V�S).

Next, let k � S0, and h � S, and consider two solutions(u3, y3) and (u4, y4) satisfying ax by � d (and, thus, �u y � ) of the following form:

(i) The star solution except that edges {0, i�} and {0, h}are removed and edges {k, i�} and {k, h} are included(if k � 0, we just take the star solution);

(ii) The previous solution except that edge {k, i�} is re-moved and edge {h, i�} is added instead.

Subtracting �u4 y4 � from �u3 y3 � yields�{k,i�} � �{h, i�} � hi� kh � ki�. By part (b), �{k,i�}

� �{h,i�} � 0, and by (26), hi� � ki� � 0. Thus, theprevious expression collapses to kh � 0. Since k and hwere chosen arbitrarily from S0 and S, respectively, weobtain kh � 0 for all arcs (k, h) � E(S0).

Finally, let k � V�S, k i�, j�. Taking i� for k, k fori, and j� for j in (5) yields i�j� � i�k � kj� � 0. But i�j�

� kj� � 0, and, thus, i�k � 0, as well. A similarargument works for j�, and, thus, we have i�k � j�k � 0for all k � V�S.

By combining the conclusions of the last few paragraphs,we conclude that hk � 0 for all arcs (h, k) � E(V�S, S).

Part (d): Show that for i � V�S, j � S, we have ij

� �. Consider two solutions (u1, y1) and (u2, y2) satisfy-ing ax by � d (and, thus, �u y � ) as follows:

(i) The star solution;(ii) The previous solution except that edge {0, j} is re-

moved and edge {i, j} is added.

By subtracting �u2 y2 � from �u1 y1 � andcanceling out the zero coefficients, we obtain � � �{0, j}

� ij, as desired. Since i and j were chosen arbitrarily fromV�S and S, we obtain ij � � for all arcs (i, j) � E(V�S,S).

Thus, we have demonstrated that the GLB inequalitiesare facet-defining for P under mild conditions on the sizesof S and V�S.

We present, now, a similar proof for the GUB inequal-ities:

�k � 1� �e�E��0��S�

ue � �i�V�S, j�S

yij � �k � 1��S� S � V.

(18)

It is not hard to see that parts (a), (b), and (c) of the previousproof are valid for the GUB inequalities, since all thecanonical solutions employed also hold at equality for theGUB constraint. The one exception is that, in part (b), thenode h should be chosen from V�S rather than arbitrarilyfrom V, which implies that the proof for the GUB inequal-ities given here is only valid for �V�S� � 3. Thus, whatremains to be shown is the relationship between �e for e� E(S0) and ij for all arcs (i, j) � E(V�S, S).

We first choose arbitrary nodes i � S and j � V�S.Consider the following pair of solutions (u1, y1) and (u2,y2):

(i) This solution contains edges {0, i}, {i, j}, and {i, k}for an additional K � 2 nodes k � V; the remainingnodes not connected to i are connected directly to theroot in star formation;

(ii) The previous solution except that edge {0, i} is re-moved and edge {0, j} is added.

By subtracting �u2 y2 � from �u1 y1 � andusing the fact that �{0,i} � ij � 0i � 0, we have�{0,i}u{0,i}

1 � jiyji2 � 0. Since yji

2 � K � 1 and u{0,i}1 �

1, it follows [upon substituting �{0,i} � �] that � � (K� 1) ji. Since j and i were chosen arbitrarily from V�S andS, respectively, the desired result follows. �

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