W. H. Cunningham, The optimal multiterminal cut problem, in DIMACS Series in Disc. Math. and Theor. Comput. Sci., vol. 5, American Mathematical Society, 1991, pp. 105--120.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and 2 trees, the general k Terminal Cut problem can be solved in linear time by a straightforward dynamic programming algorithm. This can be generalized to graphs of bounded tree width for any fixed bound, by standard techniques. The facets of the Multiterminal Cut polyhedron are studied in [2,3]. An interesting generalization of the Multiterminal Cut problem, about which we shall have more to say in our concluding section, is studied in [5,6] The 1983 abstract did not contain our proofs; these are presented here for the first time. The 1983 abstract also used the lessdescriptive term ....

....more to say in our concluding section, is studied in [5,6] The 1983 abstract did not contain our proofs; these are presented here for the first time. The 1983 abstract also used the lessdescriptive term multiway cut for what we now call a multiterminal cut. The new terminology was introduced in [3] and we adopt it here for added clarity. The paper is organized as follows. In Section 2 we cover the positive results for the planar case (Theorem 1a and 1b) The corresponding negative result for the planar case (Theorem 2) is covered in Section 3. Section 4 covers our results for general ....

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W. H. CUNNINGHAM, "The optimal multiterminal cut problem," DIMACS Series in Disc. Math. and Theor. Comput. Sci. 5 (1991), 105-120.

....edges S E whose removal disconnects each s i t i pair. This problem has been studied in the second author s PhD thesis [1] and a restricted version of the problem, called the Multiway Cut Problem (or Multiterminal Cut Problem or k Terminal Cut Problem) has also been studied quite extensively [2, 3, 5]. The Multicommodity Cut Problem has been shown to be NP hard, even if there are only three terminal pairs [5] G is a tree [15] or G is a grid graph with all terminals on the outside boundary [1] In this section we consider a further restriction on the problem. An instance of the Multicommodity ....

W.H. Cunningham, The optimal multiterminal cut problem, DIMACS Series in Disc. Math. and Theor. Comp. Sci. 5 (1991), 105--120.

....cases, far better results are known. For fixed k in planar graphs, the problem is solvable in polynomial time [9] For trees and 2 trees, there are linear time algorithms [6] For dense unweighted graphs, there is a polynomial time approximation scheme [2, 12] Chopra and Rao [6] and Cunningham [7] develop a polyhedral approach to Multiway Cut, further extended by Chopra and Owen [5] These provide useful tips to the implementation of branch andcut type heuristics that are reported by the authors to work well in practice. Bertsimas et al. 4] propose a non linear formulation of Multiway Cut ....

....cut for G in polynomial time. 4 Concluding Remarks We do not know the integrality ratio for the relaxation we proposed. It is possible that a better rounding procedure can be discovered. Here are the worst examples of which we are aware. For k = 3, the following example (which also appeared in [7]) satisfies all the new constraints and shows that the integrality ratio is at least 16 15 . Consider the graph G = V, E) V = S # 1, 2, 3 1 # S # 2 , where 1 , 2 , 3 are the terminals. E = S, T S #= T, S # T = 1 . The edges S, T with S or T of size 1 ....

W. H. Cunningham. The Optimal Multiterminal Cut Problem. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 5, pages 105--120, 1991.

....by the partition; the multiway cut consists of the edges with endnodes in di erent partition sets. The multiterminal cut problem (or multiway cut problem) arises in many applications as e.g. clustering problems and network design, and it is studied in several papers, see e.g. Chopra Rao (1991) Cunningham (1991), Johnson, Mehrotra Nemhauser (1993) In et al. 1994) it is shown that the multiterminal cut problem is NP hard for every xed r 3 when the graph is arbitrary, even when all weights are equal. They also show that the restriction of the multiterminal cut problem to planar graphs G 0 is ....

Cunningham, W. (1991), `The optimal multiterminal cut problem', DIMACS Series in Discrete Mathematics and Theoretical Computer Science 5, 105 120.

....ratio of at most 2(1 1 k) Their algorithm implements the following intuitive isolating heuristic: for each terminal, find a minimum cut separating it from all other terminals and then take the union of the k 1 cheapest minimum cuts. The multiway cut polyhedron was studied in [5, 4]. The authors identified properties of the polyhedron and in particular gave several families of facetdefining inequalities. These were used in [4] to guide a heuristic branch and cut algorithm which was reported by the authors to work well in practice. 0020 0190 00 see front matter 2000 ....

....multiway cut and related problems were studied in [2] where the authors proposed several linear relaxations solvable in polynomial time, as well as a randomized rounding technique attaining a performance guarantee of 2(1 1 k ) Results for various special cases have also been obtained. See [7,5,9,4,1,8]. The 2(1 1 k)approximation ratio due to Dahlhaus et al. 7] went unimproved for over a decade. It was improved by C alinescu et al. 3] who proposed a relaxation of the problem as an embedding of the vertices in the k simplex # k = # x # R k x # 0, # i x i = 1 # . Their ....

W.H. Cunningham, The optimal multiterminal cut problem, in: F. Roberts, F. Hwang, C. Monma (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 5, Amer. Math. Soc., Providence, RI, 1991, pp. 105--120.

....is 2. In this situation the best one can hope for is an approximate max flow min multicut theorem. We show one such by proving that the maximum flow is at least as large as the minimum multicut times Omega i 1 log k j and that there is an instance where this lower bound is achieved. Cunningham [10] showed that this gap between the maximum flow and the minimum multicut can be tightened to i 2 Gamma 2 k j for the special case when each pair of nodes, from amongst a set of k specified nodes, is the source sink pair of a commodity. The multicut now separates these k nodes and is called ....

....dynamic programming [61] The problem is also polynomial time solvable in planar graphs for fixed k; for general k it is NP hard [11] Dahlhaus et.al. 11] also give a i 2 Gamma 2 k j approximation algorithm for the minimum multiway cut problem using the idea of isolating cuts. Cunningham [10] addresses the linear programming aspects of the multiway cut problem and describes lower bounds based on these. One such lower bound is the maximum multicommodity flow obtained by allowing flow between every pair of terminals; the objective is to maximize the sum of the flows. Cunningham shows ....

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W. H. Cunningham. The optimal multiterminal cut problem. DIMACS Series in Disc. Math. and Theor. Comput. Sci., 5:105--120, 1991. 82 Bibliography 83

....more general context was that a partition into k non empty clusters minimizing the sum of distances between points in different clusters consists of k Gamma 1 singleton sets and one set of all remaining points. Exploiting the relation to cut problem, the terminal cut problem (see e.g. Cunningham [4]) could be translated into a terminal clustering, where each cluster has to contain a special terminal point. Hence, the clustering defines not only a partition of all points but also a partition of k special terminal points into k subsets. Problems of this kind occur frequently in location theory ....

W.H.Cunningham, The optimal multiterminal cut problem, DIMACS Series in Discrete Math. and Theoretical Comp. Sc. 5, 1991, 105--120.

....cases, far better results are known. For fixed k in planar graphs, the problem is solvable in polynomial time [9] For trees and 2 trees, there are linear time algorithms [6] For dense unweighted graphs, there is a polynomial time approximation scheme [2, 12] Chopra and Rao [6] and Cunningham [7] develop a polyhedral approach to Multiway Cut, further extended by Chopra and Owen [5] These provide useful tips to the implementation of branch andcut type heuristics that are reported by the authors to work well in practice. Bertsimas et al. 4] propose a non linear formulation of Multiway Cut ....

....cut for G in polynomial time. 4 Concluding Remarks We do not know the integrality ratio for the relaxation we propose. It is possible that a better rounding procedure can be discovered. Here are the worst examples of which we are aware. For k = 3, the following example (which also appeared in [7]) satisfies all the new constraints and shows that the integrality ratio is at least 16 15 . Consider the graph G = V; E) V = fS f1; 2; 3gj 1 jSj 2g, where f1g; f2g; f3g are the terminals. E = ffS; T gj S 6= T; jS T j = 1g. The edges fS; Tg with S or T of size 1 (between a terminal and ....

W. H. Cunningham. The Optimal Multiterminal Cut Problem. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 5, pages 105--120, 1991.

.... of the multiway cut problem [DJPSY] i.e. the special case of the multicut problem where the given set of source sink pairs for the commodities consists of all pairs of vertices from a given subset S of terminals, the gap between min cut and max flow is much smaller, it is at most 2 Gamma 2 k [Cu]. Of course, if S is the whole set of vertices (i.e. there is one commodity for every pair of vertices) the problem is trivial and there is no gap: max flow = min cut = total capacity of the graph. 6 Multicommodity flow with specified demands We next consider the case when along with the source ....

W. H. Cunningham (1991), "The optimal multiterminal cut problem", DIMACS Series in Disc. Math. and Theor. Comput. Sci. 5 (1991), pp. 105-120.

....and the approximation algorithm of [6] has a performance guarantee of 4 3 . Recently, Calinescu, Karloff, and Rabani [1] gave an approximation algorithm having a performance guarantee of 7 6 . We give a further improvement that is based on their approach. Chopra and Rao [3] and Cunningham [4] investigated linear programming relaxations of the 3 cut problem, showing results on classes of facets and separation algorithms. Here are the two simplest relaxations. By a T path we mean the edge set of a path joining two of the terminals. By a wye we mean the edge set of a tree having ....

....0; e 2 E: minimize P e2E c e x e (LP 2) subject to x(P ) 1; P a T path x(Y ) 2; Y a wye x e 0; e 2 E: It follows from some simple observations about shortest paths, and the equivalence of optimization and separation, that both problems can be solved in polynomial time. It was proved in [4] that the approximation algorithm of [5] delivers a 3 cut of value at most 4 3 times the optimal value of (LP 1) In particular, the minimum weight of a 3 cut is at most 4 3 times the optimal value of (LP 1) It was conjectured that the minimum weight of a 3 cut is at most 16 15 times the ....

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W.H. Cunningham, "The optimal multiterminal cut problem", in: C. Monma and F. Hwang (eds.), Reliability of Computer and Communications Networks, American Math. Soc., 1991, pp. 105--120.

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W. H. Cunningham, The optimal multiterminal cut problem, in DIMACS Series in Disc. Math. and Theor. Comput. Sci., vol. 5, American Mathematical Society, 1991, pp. 105--120.

No context found.

W. H. Cunningham. The optimal multiterminal cut problem. In Reliability of computer and communication networks (New Brunswick, NJ, 1989), volume 5 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 105--120. Amer. Math. Soc., Providence, RI, 1991.

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W. H. Cunningham, The optimal multiterminal cut problem, DIMACS Series in Disc. Math. and Theor. Comput. Sci. 5(1991) 105-120.

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