The Generalized Minimum Spanning Tree problem denoted by GMST is a variant of the classical Minimum Spanning Tree problem in which nodes are partitioned into clusters and the problem calls for a minimum cost tree spanning at least one node from each cluster. A different version of the problem, called E-GMST arises when exactly one node from each cluster has to be visited. Both GMST problem and E-GMST problem are NP-hard problems. In this paper, we model GMST problem and E-GMST problem as integer linear programs and study the facial structure of the corresponding polytopes

Pop, P.C.

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The generalized minimum spanning tree polytope and related polytopes

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Faculty of Mathematical SciencesUniversity of TwenteUniversity for Technical and Social SciencesP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlMemorandum No. 1587The generalized minimum spanningtree polytope and related polytopesP.C. PopSeptember 2001ISSN 0169-2690The Generalized Minimum SpanningTree Polytope and Related PolytopesP.C. Pop∗Faculty of Mathematical Sciences, University of TwenteP.O. Box 217, 7500 AE Enschede, The NetherlandsSeptember 12, 2001AbstractThe Generalized Minimum Spanning Tree problem denoted by GMSTis a variant of the classical Minimum Spanning Tree problem in whichnodes are partitioned into clusters and the problem calls for a minimumcost tree spanning at least one node from each cluster. A different versionof the problem, called E-GMST arises when exactly one node from eachcluster has to be visited. Both GMST problem and E-GMST problem areNP-hard problems. In this paper, we model GMST problem and E-GMSTproblem as integer linear programs and study the facial structure of thecorresponding polytopes.Keywords: combinatorial optimization, spanning trees, generalized min-imum spanning trees, facets.Mathematical Subject Classification: 90C11, 90C27, 05C05, 90B10.1 IntroductionIn this paper, we consider a generalization of the classical MST problem, calledgeneralized minimum spanning tree problem and denoted GMST, in which givena complete undirected graph G = (V,E) with the nodes partitioned into clustersand the problem is to find a minimum cost tree spanning at least one node fromeach cluster.A different version of the problem, called E-GMST problem (where E standsfor Equality), arises when imposing the additional constraint that exactly onenode from each cluster must be visited.GMST problem and E-GMST problem are useful models for problems involvingsimultaneous selection and sequencing decisions, e.g. as in location problems.∗Corresponding author. Tel: (00) 31 53 489 3385; fax: (00) 31 53 489 4858; e-mail:p.c.pop@math.utwente.nl1They find practical applications in warehouse order picking with multiple lo-cations, sequencing computer files, designing metropolitan area networks andregional area networks, settings involving agricultural irrigation systems, etc.Both GMST problem and E-GMST problem are NP-hard problems. In [9]Myung et al. proved that E-GMST problem is NP-hard, the proof being basedupon reduction from the minimum vertex cover, and in [2] Dror et al. provedthat GMST problem is NP-hard based upon reduction from the Steiner treeproblem.We now introduce the main notations used in this paper. For each S ⊆ V , letE(S) = {(i, j) ∈ E : i ∈ S, j ∈ S}δ(S) = {(i, j) ∈ E : i ∈ S, j /∈ S}µ(S) = |{k : Vk ⊆ S}|η(S) = {k : Vk ∩ S 6= φ}For i ∈ V , we write δ(i) instead of δ({i}), and denote by Vk(i) the clustercontaining i. We also defineW = {i ∈ V : |Vk(i)| = 1}2 Basic modelsLet G=(V,E) be a complete undirected graph with node set V = {1, ..., n} andedge set E = {(i, j) : i, j ∈ V, i 6= j}. In addition a proper partition V1, ..., Vm ofV is given. The node sets Vk for all k=1,...,m are called clusters. Let ce be thecost associated to each edge e ∈ E. The GMST problem is then the problemof finding a minimum-cost tree spanning a subset of nodes which includes atleast one node from each cluster. Therefore, the problem involves two relateddecisions:i) choosing a node set S ⊆ V such that |S ∩ Vk| ≥ 1 for all k=1,...,m;ii) finding a minimum cost tree in the subgraph of G induced by S.By defining the following binary variables:xij = 1 if the edge {i, j} is included in the selected subgraph0 otherwise2yi = 1 if the node i is included in the selected subgraph0 otherwisethe GMST problem can be formulated as the following integer linear program-ming:minimize∑e∈E cexesubject to y(Vk) ≥ 1, for all k=1,...,m (1)x(E(S)) ≤ y(S − i), i ∈ S ⊆ V, |S| ≥ 2 (2)x(E) ≥ y(V )− 1 (3)yi, xe ∈ {0, 1}, for all i ∈ V, e ∈ E (4)Constraints (1) force to span at least one node from each cluster. Constraints(2) are called the generalize subtour elimination constraints and they avoid thecycles. Constraints (3) imposes that the number of nodes must be greater thanthe number of edges (one may replace ≥ with = in (3)).By replacing the subtour elimination constraints with connectivity constraints,constraints (2) and (3) can be substituted byx(δ(S)) ≥ yi + yj − 1, for all S ⊂ V, i ∈ S, j /∈ S (6)x(E) ≤ y(V )− 1As to E-GMST problem, a mathematical model is obtained from (1)-(4) byreplacing (1) withy(Vk) = 1, for k=1,...,m (5)Let P, P= and Q denote, respectively the GMST, E-GMST and MST polytopes,defined asP := conv{(x, y) ∈ RE∪V : (x, y) satisfies (1)− (4)}P= := conv{(x, y) ∈ RE∪V : (x, y) satisfies (2)− (5)}Q := P ∩ {(x, y) ∈ RE∪V : yi = 1 for all i ∈ V }Clearly, P= and Q are faces of P. These faces are disjoint when m < n, whereasfor m = n, the three polytopes P, P= and Q coincide. In the next sections weanalyse the facial structure of P and P=.33 Facet-defining inequalities for the GMST poly-topeIn this section we study the GMST polytope, P. The facial structure of P isclearly related to that of the MST polytope, Q, arising when imposing theadditional equations yi = 1 for all i. To link these two polytopes, we define theintermediate polytopes:P (F ) := P ∩ {(x, y) ∈ RE∪V : yi = 1 for all i ∈ F}where φ ⊆ F ⊆ V . By definition P(V)=Q and P (φ) = P .First we determine the dimension of P(F) for any given F. In order to do thiswe study the equation system of P(F). This system includes the equations:x(E) = y(V )− 1yi = 1, for all i ∈ F ∪Wwhere W has been defined in Section 1.Theorem 1 For all F ⊆ V , dim(P (F )) = |E|+ |V | − |F ∪W | − 1.Proof. Clearly,dim(P (F )) ≤ |E|+ |V | − |F ∪W | − 1since P (F ) ⊆ RE∪V and the 1 + |F ∪W | equations describing P(F) are linearlyindependent. We claim the existence of |E|+ |V |− |F ∪W | affinely independentpoints in P(F). This will prove that:dim(P (F )) ≥ |E|+ |V | − |F ∪W | − 1and hence the theorem. The proof of the claim is by induction on the cardinalityof F.When |F | = |V | = n, the claim is true since P(F) corresponds to the spanningtree polytope.Assume now that the claim holds for |F | = σ, and consider any node set F ′with |F ′ | = σ−1. Let i /∈ F ′ and define F := F ′ ∪{i}. Because of the inductionhypothesis, there exist |E|+ |V |−|F ∪W | affinely independent points belongingto P(F) and hence to P (F′). If i ∈ W , then |F ∪ W | = |F ′ ∪ W | and wehave finished. Otherwise, |F ∪W | = |F ′ ∪W | + 1, i.e. we need an additionalpoint. Such a point exists and corresponds to any spanning tree in the subgraphinduced by V − {i}.4Corollary 2 dim(P ) = |E|+ |V | − |W | − 1.According to Theorem 1, given any nonempty F ⊆ V and any i ∈ F , one has thefollowing: if i ∈W , then dim(P (F )−{i}) = dim(P (F )), else dim(P (F )−{i}) =dim(P (F )) + 1. In other words, the removal of a node from F increases thedimension of P(F) by, at most one unit. As a consequence, any facet-defininginequality for P(F) can be lifted in a simple way so as to be facet-inducing forP (F )− {i} as well.Theorem 3 Let F ⊆ V and j ∈ F . Let∑e∈Eαexe +∑i∈Vβi(1− yi) ≥ γbe any facet-inducing inequality for P(F). Then the lifted inequality∑e∈Eαexe +∑i∈V−{j}βi(1− yi) + β̂j(1− yj) ≥ γis valid and facet-defining for P (F−{j}, where β̂j is an arbitrary value if j ∈W ,whereasβ̂j = γ −min{∑e∈Eαexe +∑i∈V−{j}βi(1− yi): (x, y) ∈ P (F − {j}) and yj = 0}holds when j /∈W .Proof. The proof of this theorem follows from the well-known sequential liftingtheorem (Padberg [9]), as described, e.g., in Grotschel and Padberg [6].Theorem 3 leads to a lifting procedure to be used to derive facet-inducing in-equalities for the GMST polytope from those of the MST polytope. To thisend, one has to choose any lifting sequence for the nodes, say {i1, ..., in}, anditeratively derive a facet of P ({it+1, ..., in}) from a facet of P ({it, ..., in}) fort=1,...,n. Different lifting sequences can produce different facets.We next use the lifting procedure to analyse the facial structure of P. We beginwith the nonnegativity constraints.Theorem 4 The inequality xe ≥ 0 defines facet of P for every e ∈ E.Proof. A direct consequence of Theorem 3, since xe ≥ 0 defines a facet of theMST polytope and every lifting sequence produces β̂i = 0 for all i ∈ V .5Notice that the inequalities yi ≥ 0 are dominated by the valid inequalitiesyi ≥ xe for e ∈ δ(i); hence they are not facet-inducing.We now analyze the upper-bound constraints and constraints (1) which force tochoose from each cluster at least one node.Theorem 5 The inequality xe ≤ 1 defines facet of P if and only if e ∈ E(W ).Proof. Let e=(i,j). If i, j ∈ W , then the claim follows from the forthcomingTheorem 8 by choosing S = {i, j}. Otherwise, xe ≤ 1 is dominated by xe ≤ yi(if i /∈W ) or xe ≤ yj (if j /∈W ).Theorem 6 The inequality yi ≤ 1 defines facet of P if and only if i /∈W .Proof. It is enough to observe that the face of P induced by yi ≤ 1 coincideswith P(F) when F := {i}; hence the claim follows from Theorem 1.Theorem 7 The inequality y(Vk) ≥ 1 does not define a facet of P for anyk=1,...m.Proof. If |Vk| = 1 then y(Vk) ≥ 1 defines an improper face of P. Otherwise,0 ≤ y(Vk)− 1 is dominated by x(E(Vk)) ≤ y(Vk)− 1 which defines a facet of Pas states in the forthcoming Theorem 8.We now study constraints (2). It is well-known that the Subtour EliminationConstraints:x(E(S)) ≤ |S| − 1are valid inequalities for the spanning tree polytope, for all S ⊆ V . It is alsoknown that they are facet-defining for all S ⊆ V such that the subgraph (S,E(S))is either 2-connected (if |S| ≥ 3) or connected (if |S| = 2) and (V −S,E(V −S))is connected. By using the lifting procedure introduced in Theorem 3 it is thenpossible to obtain new facet-defining inequalities for GMST polytope.Theorem 8 Let S be a subset of V such that the subgraph (S,E(S)) is either2-connected (if |S| ≥ 3) or connected (if |S| = 2) and let (V − S,E(V − S)) beconnected. The following Generalized Subtour Elimination Constraint is validand facet-defining for P:x(E(S)) ≤ y(S)− α(S)where6α(S) = 1 if there is a cluster Vk ⊆ Syi otherwise for every i ∈ S.Proof. We consider any lifting sequence of all the nodes V, where the nodes ofS follow all the other nodes. The inequality:x(E(S)) +∑j∈Vβj(1− yj) ≤ |S| − 1is facet-defining for the spanning tree polytope Q for any choice of βj, see Chopra[1]. We iteratively compute the lifting coefficients β̂j = 0 for the first |V − S|nodes, and β̂j = 1 for the next |S| − 1 nodes. Regarding the last node i of thesequence, we obtain β̂j = 1 if a feasible GMST solution visiting no nodes exists(i.e. no Vk ⊆ S exists), and β̂i = 0 otherwise.We finally analyse the connectivity constraints (6). Given {S1, ..., Sp} apartition of the nodes V, then the multicut constraint (or partition constraint):x(∪pi=1δ(Si)) ≥ p− 1is valid for the spanning tree polytope Q. It is also a facet-defining inequality forits dominant polyhedron if and only if the graph obtained by shrinking S1, ..., Spis 2-connected, see Chopra [1]. We can extend the above constraints to be validfor P by considering the generalized multicut contraint (or generalized partitionconstraint):x(∪pi=1δ(Si)) ≥p∑i=1α(Si)− 1where α(Si) was defined in the previous theorem.It is easy to see that the generalized multicut constraints are linear combi-nations of genaralized subtour elimination constraints and equations. Indeed,the multicut constraint is the sum of:x(E) = y(V )− 10 = α(Si)− y(Si) for all i=1,...,p such that |Si| = 1−x(E(Si)) = α(Si)− y(Si) for all i=1,...,p such that |Si| > 1.Therefore no generalized multicut constraint defines a facet of P which is notinduced by a generalized subtour elimination constraint.74 Facet-defining inequalities for the E-GMSTpolytopeWe analyse now the polyhedral structure of the E-GMST polytope, P=. Thispolytope is clearly a face of P; hence all facet-defining inequalities for P studiedin Section 3 are also valid (but not necessarily facet-defining) for P=.Since in E-GMST problem exactly one node from each cluster must be visitedwe can drop the intercluster edges and redefine the edge set asE := {(i, j) : i ∈ V, j ∈ V − Vk(i)}As in the previous section, by relating the facial structure of P= to that of thespanning tree polytope Q, even though Q is not a relaxation of P=, Feremanset al. [3] obtained the following results:Proposition 9 dim(P=) = |E|+ |V | − |K| − 1.Proof. See [3].Regarding the non-negativity constraints and the upper-bound constraints con-straints we have the following result:Proposition 10 1. The inequality xe ≥ 0 defines facet of P= for every e ∈ E.2. The inequality yi ≥ 0, i ∈ V does not define a facet of P=.3. The inequality xe ≤ 1 defines facet of P= if and only if e ∈ E(W ).Finally, by analysing the generalized subtour elimination constraints (2) we havethe following result:Proposition 11 Let S be a subset of V such that µ(S) 6= 0 and |S| ≥ 2. Thenthe generalized subtour elimination constraint (2) is facet-defining for P= if andonly if any of the following four conditions holds:i) η(S) ≥ 3 and η(V − S) ≥ 2ii) S ⊆W and |S| = 2iii) S = {i} ∪ Vk with i /∈W ∪ Vkiv) S = V − {i} with i ∈ V.Proof. See [3].8References[1] S. Chopra, On the Spanning Tree Polyhedron. Operations Research Letters,8: 25-29, 1989.[2] M. Dror, M. Haouari, J. Chaouachi, Generalized Spanning Trees EuropeanJournal of Operational Research, vol 120: 583-592, 2000.[3] C. Feremans, M. Labbe, G. Laporte, Polyhedral Analysis of the GeneralizedMinimum Spanning Tree Problem, 1999.[4] M. Fischetti, J.J. Salazar, P. Toth, The Symmetric Generalized TravellingSalesman Problem, Networks 26: 113-123, 1995.[5] M. Fischetti, J.J. Salazar, P. Toth, A Branch-and-Cut Algorithm forthe Symmetric Generalized Travelling Salesman Problem, Operations Re-search, 45: 378-394, 1997.[6] M. Grotschel and M.W. Padberg, Polyhedral Theory, in E.L. Lawler, J.K.Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys (Eds.) The TravellingSalesman Problem: A Guided Tour of Combinatorial Optimization, Wiley,New York, 1985. SIAM Journal on Applied Mathematics 9 (1961) 551-570.[7] N.K. Karmarkar, A new polynomial-time algorithm for linear program-ming. Combinatorica, 4: 373-395, 1984.[8] L. Lovasz, On some connectivity properties of Eulerian graphs. Acta Math-ematica Academiae Scientiarum Hungaricae 28 (1976) 129-138.[9] Y.S. Myung, C.H. Lee, D.w. Tcha, On the Generalized Minimum SpanningTree Problem. Networks, Vol 26 (1995) 231-241.[10] P.C. Pop, U. Faigle, W. Kern, G. Still, Relaxation Methods for the Gener-alized Minimum Spanning Tree Problem, submitted.[11] M.W. Padberg, A note on zero-one programming, Operations Research 23:833-837, 1975.[12] W.T. Tutte, On the problem of decomposing a graph into n connectedfactors. Journal of London Mathematical Society, 36: 221-230, 1961.9

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The generalized minimum spanning tree polytope and related polytopes

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