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18
The Complexity of Multiterminal Cuts
 SIAM Journal on Computing
, 1994
"... In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and ..."
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In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and can be solved in polynomial time. We show that the problem becomes NPhard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NPhard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2  2/k of the optimal cut weight.
Approximate MaxFlow Min(multi)cut Theorems and Their Applications
 SIAM Journal on Computing
, 1993
"... Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us ..."
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Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands, of LeightonRao and Klein et.al., and thereby obtain an improved bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem  the celebrated maxflow mincut theorem of Ford and Fulkerson [FF], and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graphtheoretic entities via the potent mechanism of a minmax relation. The importance of this theor...
A Sufficiently Fast Algorithm for Finding Close to Optimal Junction Trees
, 1996
"... An algorithm is developed for finding a close to optimal junction tree of a given graph G. ..."
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Cited by 77 (3 self)
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An algorithm is developed for finding a close to optimal junction tree of a given graph G.
An improved approximation algorithm for multiway cut
 Journal of Computer and System Sciences
, 1998
"... Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due ..."
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Cited by 74 (5 self)
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Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, � Johnson, Papadimitriou, Seymour, and Yannakakis gave a performance guarantee of 2 1 − 1 k. In this paper, we present a new linear programming relaxation for Multiway Cut and a new approximation algorithm based on it. The algorithm breaks the threshold of 2 for approximating Multiway Cut, achieving a. This improves the previous result for every value of k. performance ratio of at most 1.5 − 1 k In particular, for k = 3 we get a ratio of 7
Approximation algorithms for the 0extension problem
 IN PROCEEDINGS OF THE TWELFTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2001
"... In the 0extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between t ..."
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Cited by 70 (3 self)
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In the 0extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis and is closely related to the metric labeling problem introduced by Kleinberg and Tardos. We present approximation algorithms for 0Extension. In arbitrary graphs, we present a O(log k)approximation algorithm, k being the number of terminals. We also give O(1)approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem, previously considered by Karzanov. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis for the multicut problem and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least c √ lg k for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos and they further our understanding on how to use metric relaxations.
Multiway Cuts in Directed and Node Weighted Graphs
 in Proc. 21st ICALP, Lecture Notes in Computer Science 820
, 1994
"... this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for t ..."
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this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for this. Define an isolating cut for terminal s i to be a cut that separates s i from the rest of the terminals. A minimum isolating cut for s i can be computed in polynomial time by identifying the remaining terminals, and finding a minimum cut separating them from s i . The algorithm in [2] finds such cuts for each terminal, discards the heaviest cut, and picks the union of the remaining. The approximation factor is proven by observing that on doubling each edge in the optimum multiway cut, we can partition these edges into k isolating cuts, one for each Department of Computer Science and Engg., Indian Institute of Technology, New Delhi, India
Approximation algorithms for treewidth
, 2002
"... Abstract. This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in ..."
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Abstract. This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in polynomialtime and finds a factorO(log OP T), where OP T is the treewidth of the graph. This is the first polynomialtime algorithm that approximates the optimal by a factor that does not depend on n, the number of nodes in the input graph. As a result, we get an algorithm for finding pathwidth within a factor of O(log OP T · log n) from the optimal. We also present algorithms that approximate the treewidth of a graph by constant factors of 3.66, 4, and 4.5, respectively and take time that is exponential in the treewidth. These are more efficient than previously known algorithms by an exponential factor, and are of practical interest. Finding triangulations of minimum treewidth for graphs is central to many problems in computer science. Realworld problems in artificial intelligence, VLSI design and databases are efficiently solvable if we have an efficient approximation algorithm for them. Many of those applications rely on weighted graphs. We extend our results to weighted graphs and weighted treewidth, showing similar approximation results for this more general notion. We report on experimental results confirming the effectiveness of our algorithms for large
On weighted multiway cuts in trees
 MATHEMATICAL PROGRAMMING
, 1994
"... A minmax theorem is developed for the multiway cut problem of edgeweighted trees. We present a polynomial time algorithm to construct an optimal dual solution, if edge weights come in unary representation. Applications to biology also require some more complex edge weights. We describe a dynarnic ..."
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Cited by 14 (0 self)
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A minmax theorem is developed for the multiway cut problem of edgeweighted trees. We present a polynomial time algorithm to construct an optimal dual solution, if edge weights come in unary representation. Applications to biology also require some more complex edge weights. We describe a dynarnic programming type algorithm for this more general problem from biology and show that our minmax theorem does not apply to it.
Optimal 3Terminal Cuts and Linear Programming
"... . Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3cut is a subset A of E such that no two terminals are in the same component of GnA. If a nonnegative edge weight ce is specified for each e 2 E, the optimal 3cut problem is to find a 3cut of minimum total ..."
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Cited by 12 (0 self)
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. Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3cut is a subset A of E such that no two terminals are in the same component of GnA. If a nonnegative edge weight ce is specified for each e 2 E, the optimal 3cut problem is to find a 3cut of minimum total weight. This problem is NPhard, and in fact, is maxSNPhard. An approximation algorithm having performance guarantee 7 6 has recently been given by Calinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3cut has weight at most 7 6 times the optimal LP value. It is proved here that 7 6 can be improved to 12 11 , and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3cut problem having performance guarantee 12 11 . 1 Introduction Given an undirected graph G = (V; E) and k specified terminal nodes t 1 ; : : : ; t k , a kcut is a subset A of E such that no two term...
A lower bound of 8/(7 + 1/(k1)) on the integrality ratio of the CalinescuKarloffRabani relaxation for multiway cut
, 2000
"... Given an edgeweighted graph and a subset of k vertices called terminals,amultiway cut is a partition of the vertices into k components, each containing exactly one terminal. The multiway cut problem is to find a multiway cut minimizing the sum of the weights of edges with endpoints in different com ..."
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Cited by 5 (0 self)
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Given an edgeweighted graph and a subset of k vertices called terminals,amultiway cut is a partition of the vertices into k components, each containing exactly one terminal. The multiway cut problem is to find a multiway cut minimizing the sum of the weights of edges with endpoints in different components. Recently, C alinescu et al. described an approximation algorithm based on a geometric embedding of the graph's vertices into R k . We present a lower bound of 8/(7 + 1 k1 ) on the integrality ratio of this relaxation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Graph algorithms; Multiway cuts; Lower bounds; Approximation ratio 1. Introduction Let G = (V , E) be an undirected graph with nonnegative edge weights w : E # R + ,andletT ={t 1 , t 2 ,...,t k }#V be a set of k distinguished vertices called terminals.Amultiway cut in G (with respect to T ) is a partition V ={V 1 ,V 2 ,...,V k } of the vertex set such that t i # V i for all 1 # i # k....