Given a complete undirected graph with the nodes partitioned  into m node sets called clusters, the Generalized Minimum Spanning Tree problem denoted by GMST is to find a minimum-cost tree which includes exactly one node from each cluster. It is known that the GMST problem is NP-hard and even finding a near optimal solution is NP-hard. We give an approximation algorithm for the Generalized Minimum Spanning Tree problem in the case when the cluster size is bounded by $\rho$. In this case, the GMST problem can be approximated to within 2$\rho$

Kern, W.

Pop, P.C.

Still, G.J.

English

Given a complete undirected graph with the nodes partitioned into m node sets called clusters, the Generalized Minimum Spanning Tree problem denoted by GMST is to find a minimum-cost tree which includes exactly one node from each cluster. It is known that the GMST problem is NP-hard and even finding a near optimal solution is NP-hard. We give an approximation algorithm for the Generalized Minimum Spanning Tree problem in the case when the cluster size is bounded by $\rho$. In this case, the GMST problem can be approximated to within 2$\rho$

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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. 1577An approximation algorithm for thegeneralized minimum spanning treeproblem with bounded cluster sizeP.C. Pop, W. Kern and G.J. StillMarch 2001ISSN 0169-2690An Approximation Algorithm for theGeneralized Minimum Spanning TreeProblem with Bounded Cluster SizeP.C. Pop∗, W. Kern, G. StillFaculty of Mathematical Sciences, University of TwenteP.O. Box 217, 7500 AE Enschede, The NetherlandsMarch 7, 2001AbstractGiven a complete undirected graph with the nodes partitioned into mnode sets called clusters, the Generalized Minimum Spanning Tree prob-lem denoted by GMST is to find a minimum-cost tree which includesexactly one node from each cluster. It is known that the GMST problemis NP-hard and even finding a near optimal solution is NP-hard. We givean approximation algorithm for the Generalized Minimum Spanning Treeproblem in the case when the cluster size is bounded by ρ. In this case,the GMST problem can be approximated to within 2ρ.Keywords: approximation algorithms, combinatorial optimization, gen-eralized minimum spanning trees, LP relaxation.Mathematical Subject Classification: 90C11, 90C27, 05C05, 90B10.1 IntroductionThe GMST problem was introduced by Myung et al. [7]. The problem arisesin simultaneus selection and sequencing decision, e.g. as in location problems,telecommunications where metropolitan and regional area networks must be in-terconnected by a tree containing a gateway from each cluster, settings involvingagricultural irrigation systems, etc.The GMST problem is defined on an undirected graph G = (V,E) with nodespartitioned into m clusters. Let K = {1, 2, . . . ,m} be the index set of the nodesets (clusters). Then, V = V1 ∪ V2 ∪ . . . ∪ Vm and Vl ∩ Vk = ∅ for all l, k ∈ Ksuch that l 6= k. We assume that edges are defined between any two nodes∗Corresponding author. Tel: (00) 31 53 489 3385; fax: (00) 31 53 489 4858; e-mail:p.c.pop@math.utwente.nl1and each edge {i, j} ∈ E has a nonnegative cost cij . The GMST problem isthe problem of finding a minimum-cost tree spanning a subset of nodes whichincludes exactly one node from each cluster. We will call a tree containing onenode from each cluster a generalized spanning tree.The following two theorems were proved in [7].Theorem 1 The GMSTP is NP-hard. Even an approximation algorithm for the GMST problem can not be polynomial.Theorem 2 Let H be a polynomial-time heuristic for the GMST problem. As-sume P 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. However if the size of the clusters is bounded,|Vk| ≤ ρ , for all k=1,...m (a)then a polynomial approximation algorithm is possible.In this paper under condition (a) and the assumption that the costs cij are non-negative and satisfy the triangle inequality an approximation algorithm for theGMST problem with performance ratio 2ρ is given. The approximation algo-rithm is constructed following the ideas in [9] where the Generalized TravelingSalesman Problem and Group Steiner problem have been treated.2 Integer program and LP relaxation of GMSTproblemWe formulate the GMST problem as an integer programming problem. Wedefine for each edge {i, j} and each node i the binary variables:xij = 1 if edge {i, j} is included in the selected subgraph0 otherwise2yi = 1 if node i is included in the selected subgraph0 otherwiseThe GMST problem can be formulated as the following integer programmingproblem:Problem IP1:Z1 = minimize∑e∈E cexesubject to y(Vk) = 1, for all k ∈ K (1)x(δ(S)) ≥ yi, for all i ∈ S ⊂ V (2)x(E) = m− 1 (3)xe ∈ {0, 1} , for all e ∈ E (4)yi ∈ {0, 1} , for all i ∈ V (5)We use here the standard shorthand notations: for every subset S of VE(S) = {(i, j) ∈ E|i, j ∈ S}, x(E(S)) =∑e∈E(S) xe, y (S) =∑j∈S yj and asusual the cutset δ(S) is defined byδ(S) = {{i, j} ∈ E | i ∈ S and j /∈ S} .Condition (1) guarantees that a feasible solution contains exactly one vertexfrom every cluster. Condition (2) guarantees that any feasible solution is aconnected subgraph. Condition (3) simply assures that any feasible solutionhas m-1 edges and due to the fact that the cost function is non-negative thisconstraint is redundant.Consider now the LP relaxation of the integer programming formulation of theGMST problem. In order to do that, we simply replace conditions (4) and (5)in IP1 by new conditions (4′) and (5′), where(4′) 0 ≤ xe ≤ 1, for all e ∈ E(5′) 0 ≤ yi ≤ 1, for all i ∈ V3 Approximation algorithmFirst we assume that the graph has bounded cluster size, i.e. |Vk| ≤ ρ for allj=1,...,m and the cost function ce = c{i,j} is symmetric and satisfies the triangleinequality, i.e. cij + cjk ≥ cik for all i, j, k ∈ V .The algorithm for approximating the optimal solution of the GMST problem isas follows:3Algorithm ”Approximate the GMST problem”Input: A complete graph G=(V,E) with non-negative symmetric cost func-tion on the edges satisfying the triangle inequality, and clusters V1, ..., Vm suchthat V = V1 ∪ V2 ∪ . . . ∪ Vm and with bounded size.Output: A tree T ⊂ G spanning some vertices W ′ ⊂ V which includesexactly one vertex from every cluster, that approximates the optimal solutionto the GMST problem.1. Solve the LP relaxation of the problem IP1 and let(y∗, x∗, Z∗1 ) = ((y∗i )ni=1, (x∗e)e∈E , Z∗1 ) be the optimal solution.2. Set W ∗ ={i ∈ V |y∗i ≥ 1ρ}and consider W′ ⊂W ∗ with the property thatW′has exactly one vertex from each cluster, and find a minimum spanning treeT ⊂ G on the subgraph G′ generated by W ′ .3. Output APP = length(T) and the tree T.Even though the LP relaxation of the problem IP1 has exponentially manyconstraints, it can still be solved in polynomial time either using ellipsoid methodwith a min-cut max-flow oracle [4] or using Karmakar’s algorithm [5] since theLP relaxation can be formulated compactly using flow variables [7, 8].4 Auxiliary resultsIn order to stablish upper bounds on the performance ratio of the above algo-rithm, we now present some auxiliary results.The parsimonius property proven by Goemans and Bertsimas [3] can be statedas follows: considering a complete undirected graph G=(V,E), for any pair(i,j) of vertices, let rij be the connectivity requirement between i and j (rij isassumed to be symmetric,i.e. rij = rji). Consider now the following two integerprograms:IZφ(r) = min∑e∈E cexes.t. x(δ(S)) ≥ max(i,j)∈δ(S)rij , for all S ⊂ V , S 6= φ0 ≤ xe, for all e ∈ Exe integral, for all e ∈ ELet denote Zφ(r) the optimal value of the LP relaxation (obtained by droppingthe integrality restrictions). Clearly Zφ(r) is a lower bound on IZφ(r).4IZD(r) = min∑e∈E cexes.t. x(δ(S)) ≥ max(i,j)∈δ(S)rij , for all S ⊂ V , S 6= φx(δ(i)) = maxj∈V \{i}rij , i ∈ D, D ⊆ V0 ≤ xe, for all e ∈ Exe integral, for all e ∈ ELet ZD(r) the optimal value of the LP relaxation.Theorem 3 (parsimonious property) If the costs cij satisfy the triangle inequal-ity, thenZφ(r) = ZD(r)for all subsets D ⊆ V. The proof of this theorem is based on a result due to Lovasz [6] on connectivityproperties of Eulerian multigraphs.Let W ⊂ V . Consider the following linear program.Problem LP2:Z∗2 (W ) = min∑e∈E cexes.t. x(δ(S)) ≥ 1, for all S ⊂ V s.t. W ∩ S 6= φ 6= W \ S (6)∑e∈δ(i) xe = 0, for all i ∈ V \W (7)0 ≤ xe ≤ 1, for all e ∈ E (8)Let us consider the following relaxation of problem LP2.Problem LP3:Z∗3 (W ) = min∑e∈E cexes.t. x(δ(S)) ≥ 1, for all S ⊂ V s.t. W ∩ S 6= φ 6= W \ S (6)0 ≤ xe, for all e ∈ E (9)Thus we omitted constraint (7) and relaxed constraint (8).The following result is a straightforward consequence of the parsimonius prop-erty, if we choose rij=1 if i, j ∈ S and 0 otherwise, and D=V \W .Lemma 4 The optimal solution values to problems LP2 and LP3 are the same,that isZ∗2 (W ) = Z∗3 (W ).5Consider the following problem:Problem IP4:Z4 = minimize∑e∈E cexesubject to x(δ(S)) ≥ 1, for all S ⊂ V , φ 6= S 6= V (10)xe ∈ {0, 1} , for all e ∈ E (11)Clearly, it is the integer programming formulation of the MST (minimum span-ning tree) problem. Let LP4 be the LP relaxation of this formulation, that iswe simply replace the constraint (11) by a new constraint (11′), where(11′) 0 ≤ xe ≤ 1, for all e ∈ EDenote by Z∗4 the value of the optimal solution of the LP4.Proposition 5LT (V ) ≤ (2− 2|V | )Z∗4 .Proof. Let S be a proper nonempty subset of V. Denote by S the complementof S in V that is S = V \ S. Constraints (10) imply that for all i ∈ V1 ≤∑e∈δ(i)xeand summing over all i ∈ S, we get|S| ≤∑i∈S∑e∈δ(i)xe = 2∑e∈E(S)xe +∑e∈δ(S)xeAdding constraint (10) gives|S|+ 1 ≤∑i∈S∑e∈δ(i)xe +∑e∈δ(S)xe = 2∑e∈E(S)xe + 2∑e∈δ(S)xeNow, it easy to see that|S|+ 1 = |S|+ 1|S||S| ≥ |V ||V | − 1 |S| =|V ||V | − 1(|V | − |S|).Therefore the problem:6min∑e∈E cexes.t.∑e/∈E(S) xe ≥|V |2(|V |−1)(|V | − |S|), for all S ⊂ V , φ 6= S 6= V (12)xe ≥ 0, for all e ∈ E (13)is a valid relaxation of the LP4 problem. Combining the results of Fulkerson [2],Edmonds [1] and Tutte [10], we see that the extremal points of the polyhedradetermined by constraints (12) and (13) are |V |2(|V |−1) multiples of spanning trees.Therefore, the length of the minimum spanning tree on V is no longer than(2− 2|V | )Z∗4 .Let W ⊂ V . We can easily modify Proposition 5 to obtain:Proposition 6LT (W ) ≤ (2− 2|W | )Z∗2 (W ).Proof. Since for any feasible solution to LP2, e /∈ E(W ) implies xe = 0, wecan use Proposition 5 to prove the inequality.5 Performance BoundsLet (y∗, x∗, Z∗1 ) = ((y∗i )ni=1, (x∗e)e∈E , Z∗1 ) be the optimal solution to the LP re-laxation fo the GMST problem. Definex̂e = ρx∗eŷi =1 if y∗i ≥ 1ρ0 otherwiseW ∗ ={i ∈ V |y∗i ≥ 1ρ}= {i ∈ V |ŷi = 1}. Because we need only one vertex fromevery cluster we delete extra vertices from W ∗ and consider W′ ⊂W ∗ such that|W ′ | = m and W ′ consists of exactly one vertex from every cluster.Since LP1 is the LP relaxation of the problem IP1, we haveZ∗1 ≤ Z1Now let us show that (x̂e)e∈E is a feasible solution to LP3. Indeed, x̂e ≥ 0 forall e ∈ E, hence condition (9) is satisfied. Let S ⊂ V be such that W ′ ∩ S 6=7φ 6= W ′ \ S and choose some i ∈ W ′ ∩ S. Hence ŷi = 1 and y∗i ≥ 1ρ . Then wehave ∑e∈δ(S)x̂e = ρ∑e∈δ(S)x∗e ≥ ρy∗i ≥ ρ1ρ= 1by definition of x̂e and the fact that the x∗e are solution to LP1. Hence the x̂esatisfy constraint (6) in LP3.Therefore,APP = LT (W′) ≤ (2− 2|W ′ | )Z∗2 (W′) = (2− 2|W ′ | )Z∗3 (W′)≤ (2− 2|W ′ | )∑e∈Ecex̂e = (2−2|W ′ | )ρ∑e∈Ecex∗e = (2−2|W ′ | )ρZ∗1≤ (2− 2|W ′ | )ρZ1 = (2−2|W ′ | )ρOPT.And since W′ ⊂ V , that is m = |W ′ | ≤ |V | = n, we have proved the following.Theorem 7 The performance ratio of the algorithm ”Approximate GMST prob-lem” for approximating the optimum solution to the GMST problem satisfies:APPOPT≤ (2− 2n)ρ.Note: One can easily generalize the algorithm and its analysis to the casewhen, in addition to distances between edges, there is a cost associated witheach vertex. In fact, we can show that, in this case,APP = LAPP + CAPP ≤ ρ(2−2n)LOPT + ρCOPT≤ ρ(2− 2n)(LOPT + COPT ) ≤ ρ(2−2n)OPT.References[1] J. Edmonds, Minimum partition of a matroid into independent sets. Journalof Research of the National Bureau of Standards B, 69B: 67-72, 1965.[2] D.R. Fulkerson, Blocking and anti-bloking polyhedra. Mathematical Pro-gramming, 1(2): 168-194, Nov. 1971.[3] M.X. Goemans and D.J. Bertsimas, Surviveble networks, linear program-ming relaxations and the parsimonious property. Mathematical Program-ming, 60: 145-166, 1993.8[4] R.E. Gomory and T.C. Hu, Synthesis of a communication network. SIAMJournal on Applied Mathematics 9 (1961) 551-570.[5] N.K. Karmarkar, A new polynomial-time algorithm for linear program-ming. Combinatorica, 4: 373-395, 1984.[6] L. Lovasz, On some connectivity properties of Eulerian graphs. Acta Math-ematica Academiae Scientiarum Hungaricae 28 (1976) 129-138.[7] Y.S. Myung, C.H. Lee, D.w. Tcha, On the Generalized Minimum SpanningTree Problem. Networks, Vol 26 (1995) 231-241.[8] P.C. Pop, U. Faigle, W. Kern, G. Still, Relaxation Methods for the Gener-alized Minimum Spanning Tree Problem, submitted.[9] P. Slavik, On the Approximation of the Generalized Traveling SalesmanProblem, submitted.[10] W.T. Tutte, On the problem of decomposing a graph into n connectedfactors. Journal of London Mathematical Society, 36: 221-230, 1961.9

An approximation algorithm for the generalized minimum spanning tree problem with bounded cluster size

NARCIS 

A new polynomial-time algorithm for linear programming.

Blocking and anti-bloking polyhedra.

Minimum partition of a matroid into independent sets.

On some connectivity properties of Eulerian graphs.

On the Approximation of the Generalized Traveling Salesman Problem,

On the Generalized Minimum Spanning Tree Problem.

On the problem of decomposing a graph into n connected factors.

Relaxation Methods for the Generalized Minimum Spanning Tree Problem,

Surviveble networks, linear programming relaxations and the parsimonious property.

http://purl.utwente.nl/publications/65764

An approximation algorithm for the generalized minimum spanning tree problem with bounded cluster size

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