We consider the Generalized Minimum Spanning Tree Problem denoted by GMSTP. It is known that GMSTP is NP-hard and even finding a near optimal solution is NP-hard. We introduce a new mixed integer programming formulation of the problem which contains a polynomial number of constraints and a polynomial number of variables. Based on this formulation we give an heuristic solution, a lower bound procedure and an upper bound procedure and present the advantages of our approach in comparison with an earlier method. We present a solution procedure for solving GMST problem using cutting planes

Kern, W.

Pop, P.C.

Still, G.J.

English

Kern, Walter

Still, Georg J.

University of Twente Research Information

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. 1542The generalized minimum spanning tree problemP.C. Pop, W. Kern and G.J. StillSeptember 2000ISSN 0169-2690The Generalized Minimum Spanning TreeProblemP.C. Pop, W. Kern, G. StillThe University of TwenteSeptember 18, 2000AbstractWe consider the Generalized Minimum Spanning Tree Problem de-noted by GMSTP. It is known that GMSTP is NP-hard and even findinga near optimal solution is NP-hard. We introduce a new mixed integerprogramming formulation of the problem which contains a polynomialnumber of constraints and a polynomial number of variables. Based onthis formulation we give an heuristic solution, a lower bound procedureand an upper bound procedure and present the advantages of our approachin comparison with an earlier method. We present a solution procedurefor solving GMST problem using cutting planes.Keywords: generalized minimum spanning trees, lower and upperbounds, cutting planes.Mathematical Subject Classification: 90C11, 90C35.1 IntroductionThis paper is concerned with the generalized version of the minimum spanningtree problem (MSTP) called the generalized minimum spanning tree problem(GMSTP). Given an undirected graph whose nodes are partitioned into mutu-ally exclusive and exhaustive node sets, the GMSTP is then to find a minimum-cost tree which includes exactly one node from each set. Therefore, the MSTPis a special case of the GMSTP where each node set consists of exactly onenode.The GMSTP has been introduced in [8]. It has many applications in thereal world. The GMSTP model fits various problems of determining the lo-cation of regional service centers (e.g. public facilities, branches, distributioncenters) which should be connected by establishing or building links (e.g. high-ways, communication links). For example, when a company tries to establishmarketing centers, one for each partitioned market, and construct a communi-cation network which interconnects the established centers, the company facesa GMSTP. For another example, when designing metropolitan area networks1[2] and regional area networks [7], we are to interconnect a number of local areanetworks. For this internetworking, we must select a node in each local networkas a hub (or a gateway) and connect the hub nodes via transmission links suchas optical fibers. Then, such a network design problem reduces to a GMSTP.2 Definition and complexity of the problemThe GMSTP is defined on an undirected graph G = (V,E) with nodes parti-tioned intommutually exclusive and exhaustive node sets. LetK = {1, 2, . . . ,m}be the index set of the node sets (clusters). Then, V = V1 ∪ V2 ∪ . . . ∪ Vm andVl ∩ Vk = ∅ for all l, k ∈ K such that l 6= k. We assume that edges are definedonly between nodes belonging to different sets and each edge {i, j} ∈ E has anonnegative cost cij . The GMSTP is the problem of finding a minimum-costtree spanning a subset of nodes which includes exactly one node from each nodeset. We will call a spanning tree containing one node from each node set as ageneralized spanning tree.Although the GMSTP generalizes the MSTP, it is an NP-hard problem.Theorem 1 The GMSTP is NP-hard.It can be shown that even finding a near optimal solution for the GMSTPis an NP-hard problem.Theorem 2 Let H be a polynomial-time heuristic for the GMSTP. AssumeP 6= NP . Then no value L ∞ can exist such thatZH (I)Z (I)≤ Lfor every instance I where Z (I) and ZH (I) are the values of an optimal solutionand of the solution found by H, respectively.In order to prove these two theorems Myung et al [6] considered the node-cover problem which is NP-complete [1]. Given a graph G = (V,E) and aninteger k   |V |, the node-cover problem is to determine whether a graph has aset C of at most k nodes such that all edges of G are adjacent to at least onenode of C. We call such a set C a node-cover of G. They proved that, givenG = (V,E) and k one can construct a graph G′ = (V ′, E′) such that G containsC if and only if there exists a generalized spanning tree in G′.23 Model formulationsWe are going to give a formulation of the GMSTP as a programming problem.We define for each edge {i, j} and each node i the binary variables:xij = 1 if the edge {i, j} is included in the selected subgraph0 otherwisezi = 1 if the node i is included in the selected subgraph0 otherwiseA feasible solution of the GMSTP can be seen as a minimal subgraph havingm − 1 edges, one node selected from every cluster and containing no cycles.Therefore, the GMSTP can be formulated as the following 0-1 mixed integerprogramming problem:minimize∑{i,j}∈E cijxijsubject to z(Vk) = 1 k ∈ K (1)x(E(S)) ≤ z(S − i) ,i ∈ S ⊂ V (2)x(E) = |K| − 1 (3)zi, xij ∈ {0, 1} ,i ∈ V, {i, j} ∈ E (4)where S − i = S\ {i} and z (S − i) =∑j∈S−i zj .This formulation is called the generalized subtour elimination formulationsince it contains constraints (2) that eliminate all the subtours (cycles).We denote the feasible set of the linear programming relaxation of thisformulation by Psub , where we replace the constraints xij , zi ∈ {0, 1} with0 ≤ xij , zi ≤ 1.Given a subset S of V ,we define the cutset δ(S) byδ(S) = {{i, j} ∈ E | i ∈ S and j /∈ S}A feasible solution of the GMSTP can also be seen as a minimal subgraphhaving m − 1 edges, one node selected from every cluster and having a pathbetween any pair of selected nodes from different clusters, therefore we canformulate the GMSTP as the following 0-1 mixed integer programming problem:minimize∑{i,j}∈E cijxijsubject to (1), (3), (4) andx(δ(S)) ≥ zi + zj − 1 ,i ∈ S ⊂ V and j /∈ S (5)3This formulation is called the generalized cutset formulation since it con-tains constraints (5) that guarantee the existence of a path between any pair ofselected nodes from different clusters. We denote the feasible set of its linearprogramming relaxation by Pcut.Theorem 3 The following properties hold.a) We have Psub ⊂ Pcut.b) The polyhedron Pcut may have fractional extreme points.Proof. a)(See also [8])Let (x, z) ∈ Psub and i ∈ S ⊂ V and j /∈ S.For any subset S of nodes we have: E = E(S)∪ δ(S)∪E(V \ S). Thereforex(δ(S)) = x(E) − x(E(S)) − x(E(V \S))≥ z(V )− 1− z(S) + zi − z(V \S) + zj= zi + zj − 1For the second part of the theorem it is easy to give an example.Myung et al tried to solve the GMSTP with the help of the generalized min-imum spanning arborescence problem on a directed graph D = (V,A) denotedby GMSAP. The GMSTP on a graph G = (V,E) can be transformed into GM-SAP by bidirecting every edge of G, choosing arbitrarily a root node set V1, andletting the costs of both arcs (i, j) and (j, i) be the same as the cost of edge{i, j} .The GMSAP can be formulated as the following mixed integer programmingproblem using multicommodity flow variables fkij (see [8,§3.2]):mimimize∑{i,j}∈A cijxijsubject to z(Vk) = 1, k ∈ Kx(δ−(Vk)) ≤ 1, k ∈ K1fk(δ+ (i))− fk(δ− (i)) = zi, i ∈ V1−zi, i ∈ Vk0, i /∈ V1 ∪ Vk , k ∈ K1fkij ≤ xij , (i, j) ∈ A, k ∈ K1fkij ≥ 0 (i, j) ∈ A, k ∈ K1zi, xij ∈ {0, 1} (i, j) ∈ A, i ∈ V.Here fkij represents the amount of flow leaving the root node V1 going via i,jand entering the cluster Vk, V1 is the root node, K1 = K\ {1}, x is the incidencevector of the selected arborescence.This formulation has a polynomial number of constraints and a polynomialnumber of variables. By using the nice structure of the dual LP relaxationMyung [6] gave an efficient algorithm for solving GMSTP.44 A new formulation of the problemIn this section we give another formulation for the GMSTP. We shrink all thevertices from each cluster in one. Our new formulation aims at distinguishingbetween global, i.e., inter-cluster connections, and local ones. We introducevariables yij (i, j ∈ {1, ...,m}) to describe the global connections. So yij = 1 ifcluster Vi is connected to cluster Vj and yij = 0 otherwise. The convex hullof all these y-vectors is generally known as the spanning tree polytope (on thecontracted graph with vertex set {V1, ..., Vm} which we assume to be complete).Following Yannakakis [9] this polytope can be represented by the followingpolynomial number of constraints:∑{i,j} yij = m− 1yij = λkij + λkji, for 1 ≤ k, i, j ≤ m and i 6= j (6)∑j λkij = 1, for 1 ≤ k, i, j ≤ m and i 6= k (7)λkkj = 0, for 1 ≤ k, j ≤ m (8)yij , λkij ≥ 0, for 1 ≤ k, i, j ≤ mwhere the variables λkij is defined for every triple of nodes k, i, j, withi 6= j 6= k and its value for a spanning tree is:λkij = 1 if j is the parent of i when we root the tree at k0 otherwiseThe constraints (6) mean that an edge {i, j} is in the spanning tree if andonly if either i is the parent of j or j is the parent of i; the constraints (7) meanthat if we root a spanning tree at k then every node has a parent and finallyconstraints (8) mean that the root k has no parent.If the 0-1 vector y describes a spanning tree on the contracted graph, thecorresponding ”local solution” x ∈ {0, 1}|E| can be obtained by solving thefollowing 0-1 mixed integer programming problem:c(y)=minimize∑{i,j}∈E cijxijsubject to z (Vk) = 1, k ∈ K (9)x(Vl, Vr) = ylr 1 ≤ l, r ≤ m (10)x (i, Vr) ≤ zi r ∈ K, i ∈ V, i /∈ Vr (11)x(E) = m− 1 (12)xij , zi ∈ {0, 1} {i, j} ∈ E, i ∈ Vwhere x(Vl, Vr) =∑i∈Vl,j∈Vr xij and x (i, Vr) =∑j∈Vr xij .We denote the feasible set of the linear programming relaxation of this prob-lem by Plocal, where we replace the constraints xij , zi ∈ {0, 1}with 0 ≤ xij , zi ≤1 and the matrix associated with the constraints (9)-(12) by A. Therefore5Plocal = {(x, z) |A (x z) ≤ b, 0 ≤ xij , zi ≤ 1}Note that the vector b is integral.Now we are able to formulate the GMSTP as an 0-1 mixed integer program-ming problem with a polynomial number of variables and a polynomial numberof constraints:minimize∑{i,j}∈E cijxijsubject to z (Vk) = 1, k ∈ K (13)x (E) = m− 1 (14)x (Vl, Vr) = ylr 1 ≤ l, r ≤ m (15)x(i, Vr) ≤ zi r ∈ K, i ∈ V, i /∈ Vr (16)(P0)ylr = λklr + λkrl 1 ≤ k, l, r ≤ m with l 6= r∑r λklr = 1 1 ≤ k, l, r ≤ m with k 6= lλkkr = 0 1 ≤ k, r ≤ m with k 6= r (17)ylr, λklr ≥ 0 1 ≤ k, l, r ≤ mxij , zi ∈ {0, 1} {i, j} ∈ E, i ∈ V (18)We denote the convex hull of the projections of all solutions to (13)-(17) and0 ≤ xij , zi ≤ 1 onto the space of x, z variables by PGMSTP ,PGMSTP = conv{(x, z) | there exist ylr, λklr such that (13)-(17) holdand 0 ≤ xij , zi ≤ 1}The following result can be proven:Proposition 4 Let {i, j} ∈ E, i, j ∈ V, r ∈ K. Then each of the followinginequalities defines a facet of PGMSTP :i) xij ≥ 0;ii) xij − zi ≤ 0 and xij − zj ≤ 0;iii) zi + zj − xij − x (i, Vr)− x (j, Vr) ≤ 0.5 Solution procedure and computational resultsbased on the new formulationThere are different ways to solve GMSTP with the help of formulation (P0).The first possibility is to solve the mixed integer program (P0) directly (forexample with CPLEX).Secondly we considered the relaxation of (P0) obtained by choosing randomlyone cluster Vk and rooting the tree only at the root k.6minimize∑{i,j}∈E cijxijsubject to z (Vk) = 1, k ∈ Kx (E) = m− 1x (Vl, Vr) = ylr, 1 ≤ l, r ≤ mx (i, Vr) ≤ zi, r ∈ K, i ∈ V, i /∈ Vr(Pk0) ylr = λklr + λkrl, 1 ≤ k, l, r ≤ m with l 6= r, k fixed∑r λklr = 1, 1 ≤ k, l, r ≤ m with k 6= l, k fixedλkkr = 0, 1 ≤ k, r ≤ m with k 6= r, k fixedylr, λklr ≥ 0, 1 ≤ k, l, r ≤ m with k fixedxij , zi ∈ {0, 1} , {i, j} ∈ E, i ∈ V.If the optimal solution of this relaxation produces a generalized spanningtree, then we have given the optimal solution of GMSTP. Otherwise we couldchoose another root or add a second root.Furthermore we considered the LP-relaxation of (Pk0). This relaxation leadsto lower bounds for the GMSTP which are comparable with the lower boundsgiven in [6], but can be computed faster.In order to strengthen the lower bounds for the GMSTP one can add validinequalities given in Proposition 4 iii).To obtain upper bounds we compute a solution ylr of the minimum spanningtree problem:minimize∑dlrylrsubject to y ∈ P0where dlr = min {cij |i ∈ Vl, j ∈ Vr} ; l, r ∈ K.With these global values ylr we solve the 0-1 linear programming problem:minimize∑{i,j}∈E cijxijsubject to z (Vk) = 1, k ∈ Kx (E) = m− 1x (Vl, Vr) = ylr 1 ≤ l, r ≤ mx (i, Vr) ≤ zi 1 ≤ r ≤ m, i ∈ V, i /∈ Vr0 ≤ xij , zi ≤ 1 {i, j} ∈ E, i ∈ V.Obviously, this leads to upper bounds for the GMSTP.Our algorithms have been coded in C and for solving the linear and mixedinteger programming problems we used CPLEX. For the test problems that weused, the edge costs were randomly generated on the [0,100] interval.Table 1 shows the computational results for solving GMSTP directly via(P0) and the relaxation of GMSTP obtained by rooting the tree at one randomlygenerated root k, for some randomly generated data instances.7Problem size GMSTP Relaxation of GMSTPm n8 4610 4612 4615 4818 520 325 34CPU(s) LP-CPU0.73 0.321.01 0.674.54 1.434.70 3.066.25 4.7312.73 9.016.25 4.7375.57 22.72- -- -- -- -CPU(s) LB/OPT(%)0.10 1000.14 1000.13 840.25 1000.21 1000.57 1000.73 1003.96 1002.76 1000.78 1001.70 1002.97 100The first column in the table gives the number of node sets (m) and thenumber of nodes per set (n). The second column shows the computation timein seconds for solving the GMSTP and the LP-relaxation of GMSTP by CPLEX.Finally the last column shows the computation time for solving the relaxation ofGMSTP obtained by rooting the tree at root k and the lower bounds attainedas a percentage of the optimal objective value. As it can be seen in almostall the cases considered the average LB/Opt is 100,i.e. the optimal solution ofGMSTP has been found. In all the examples when the first chosen root didnot lead to the optimum of GMSTP (i.e. LB/OPT  100) a second choice fora root solved the generalized minimum spanning tree problem to optimality.These first numerical experiences with the new formulation (P0) are promising.The computational results so far could indicate that the exponential growthof the computation time for the relaxation in the third column is moderate incomparison with the growth of computation time for solving the full problemdirectly (see column 2).6 Solving GMST problem by cutting planesDue to the exponential number of constraints (2) of the generalized subtourelimination formulation of the GMST problem, it would be unwise to attemptto solve the GMST problem by making use of all the generalized subtour elim-ination constraints (only few of these need to be explicitly used to obtain anoptimal solution to the GMST problem).We consider the following relaxation of the subtour elimination formulationof the GMST problem:8minimize∑{i,j}∈E cijxijsubject to z(Vk) = 1 k ∈ K(P1) x(E(S)) ≤ z(S − i) ,i ∈ S ⊂ V , |S| = 2x(E) = |K| − 1zi, xij ∈ {0, 1} ,i ∈ V, {i, j} ∈ EIf an optimal solution of this program implies a generalized spanning treethen it also must be an optimal solution for the GMST problem.Whenever this is not the case, one or more inequalities (2) for 3 ≤ |S| ≤|V | − 1 are violated by this solution. One can append violated inequalities andreoptimize the resulting program.Proceeding in this way one would solve the GMST problem.A crucial step in such a cutting plane procedure is the separation problemfor the subtour elimination constraints (2): given a point x find an inequalityof type (2) that is violated by x or prove that no such inequality exists.We present some numerical experiments. Again the costs cij between verticesi, j ∈ V from different clusters have been generated randomly in the interval[0,100]. In order to solve the integer programming problems we used CPLEX.To check whether an optimal solution of the relaxation of the subtour elimi-nation formulation (P1) implies a generalized spanning tree we have used anO(|K||V |) algorithm which detects the cycles. If there is no cycle then we gotthe solution of the GMST problem otherwise we detect the cycles and add to(P1) the corresponding violated constraints.Table 2 shows the computational results.Problem size No. of iterations No. of constraints CPU time (s)10 4 4 15 1.2715 4 2 6 2.0925 3 3 11 2.75The first two entries contain the size of the problem: the number of clustersand the number of nodes per cluster. The third entry gives the number of mixedinteger programming problems solved before optimality was proven. The forthentry gives the total number of generalized subtour elimination constraints thatwere generated during the solution procedure and appended to the (P1). Finally,the fifth column gives the CPU time in order to find an optimal solution of theGMST problem.It can be shown that also the separation problem for the subtour eliminationconstraints of the LP relaxation of (P1) can be solved in O(|V |4) time by solvinga sequence of maximum flow problems in an appropriately defined directednetwork [6]. This can be done in a similar way as proposed in [5] for theMinimum Spanning Tree problem and Steiner Tree problem.9References[1] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide tothe Theory of NP-completeness. Freeman, San Francisco (1979)[2] M. Gerla and L. Frata, Tree structured fiber optics MAN’s. IEEE J.Select.Areas Comm. SAC-6 (1988), 934-943.[3] A. Ghouila-Houri, Caracterisation des matrices totalement unimodulaires,Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences(Paris) 254, (1962) 1192-1194.[4] A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex poly-hedra, in: Linear Inequalities and Related Systems (H.W.Kuhn and A.W.Tucker, eds.), Princeton Univ. Press, 1956, pp. 223-246.[5] A. Lucena and J.E. Beasley, Branch and cut algorithms, in Advances inLinear and Integer Programming, Oxford Lecture Series in Mathematicsand its Applications, 1996.[6] M.W. Padberg and L.A. Wolsey, Trees and cuts, in Ann. Discr. Math. 17(1983), 511-517.[7] J.J. Prisco, Fiber optic regional area networks in New York and Dallas. IEEEJ. Select. Areas Comm. SAC-4 (1986), 750-757.[8] Y.S. Myung, C.H. Lee, D.w. Tcha, On the Generalized Minimum SpanningTree Problem. Networks, Vol 26 (1995) 231-241.[9] M. Yannakakis, Expressing combinatorial optimization problems by linearprograms. Journal of Computer and System Sciences 43, 441-466 (1991).10

The generalized minimum spanning tree problem

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