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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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Cited by 190 (0 self)
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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.
Cluster analysis and mathematical programming
 MATHEMATICAL PROGRAMMING
, 1997
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Globally Optimal Image Partitioning by Multicuts
"... We introduce an approach to both image labeling and unsupervised image partitioning as different instances of the multicut problem, together with an algorithm returning globally optimal solutions. For image labeling, the approach provides a valid alternative. For unsupervised image partitioning, the ..."
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Cited by 20 (10 self)
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We introduce an approach to both image labeling and unsupervised image partitioning as different instances of the multicut problem, together with an algorithm returning globally optimal solutions. For image labeling, the approach provides a valid alternative. For unsupervised image partitioning, the approach outperforms stateoftheart labeling methods with respect to both optimality and runtime, and additionally returns competitive performance measures for the Berkeley Segmentation Dataset as reported in the literature.
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....
Twoconnected Augmentation Problems in Planar Graphs
, 1996
"... Given an weighted undirected graph G and a subgraph S of G, we consider the problem of adding a minimumweight set of edges of G to S so that the resulting subgraph satisfies specified (edge or vertex) connectivity requirements between pairs of nodes of S. This has important applications in upgr ..."
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Given an weighted undirected graph G and a subgraph S of G, we consider the problem of adding a minimumweight set of edges of G to S so that the resulting subgraph satisfies specified (edge or vertex) connectivity requirements between pairs of nodes of S. This has important applications in upgrading telecommunications networks to be invulnerable to link or node failures. We give a polynomial algorithm for this problem when S is connected, nodes are required to be at most 2connected, and G is planar. Applications to network design and multicommodity cut problems are also discussed.
Analysis of LP relaxations for multiway and multicut problems
 M.C. COSTA ET AL. / EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
, 1999
"... We introduce in this paper an exact nonlinear formulation of the multiway cut problem. By simple linearizations of this formulation, we derive several wellknown and new formulations for the problem. We further establish a connection between the multiway cut and the maximumweighted independent set ..."
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We introduce in this paper an exact nonlinear formulation of the multiway cut problem. By simple linearizations of this formulation, we derive several wellknown and new formulations for the problem. We further establish a connection between the multiway cut and the maximumweighted independent set problem. This leads to the study of several instances of the multiway cut problem through the theory of perfect graphs. We also introduce a new randomized rounding argument to study the
Approximability of the kServer Disconnection Problem
"... Consider a network of k servers and their users. Each server provides a unique service that has a certain utility for each user. Now comes an attacker who wishes to destroy a set of network edges to maximize his net gain, namely the total disconnected utilities of the usersminus the total edgedest ..."
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Consider a network of k servers and their users. Each server provides a unique service that has a certain utility for each user. Now comes an attacker who wishes to destroy a set of network edges to maximize his net gain, namely the total disconnected utilities of the usersminus the total edgedestruction cost. This kserver disconnection problem is NPhard and, furthermore, cannot be approximated within a polynomially computable factor of the optimumwhen k is part of the input. Even for anyfixed k ≥ 2, there is a constant > 0 such that approximation of the problem within a factor 1/(1 + ) of the optimum is NPhard. However, a
Nonlinear formations and improved randomized algorithms for multiway and multicut problems
, 1995
"... We introduce nonlinear formulations of the multiway cut and multicut problems. By simple linearizations of these formulations we derive several well known formulations and valid inequalities as well as several new ones. Through these formulations we establish a connection between the multiway cut an ..."
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We introduce nonlinear formulations of the multiway cut and multicut problems. By simple linearizations of these formulations we derive several well known formulations and valid inequalities as well as several new ones. Through these formulations we establish a connection between the multiway cut and the maximum weighted independent set problem that leads to the study of the tightness of several LP formulations for the multiway cut problem through the theory of perfect graphs. We also introduce a new randomized rounding argument to study the worst case bound of these formulations, obtaining a new bound of 2a(H)(1) for the multicut problem, where ac(H) is the size of a maximum independent set in the demand graph H. 1