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22
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.
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.
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.
Fast Approximation Algorithms for Cutbased Problems in Undirected Graphs
"... We present a general method of designing fast approximation algorithms for cutbased minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs wh ..."
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Cited by 20 (3 self)
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We present a general method of designing fast approximation algorithms for cutbased minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs while only losing a polylogarithmic factor in the approximation guarantee. To illustrate the applicability of our paradigm, we focus our attention on the undirected sparsest cut problem with general demands and the balanced separator problem. By a simple use of our framework, we obtain polylogarithmic approximation algorithms for these problems that run in time close to linear. The main tool behind our result is an efficient procedure that decomposes general graphs into simpler ones while approximately preserving the cutflow structure. This decomposition is inspired by the cutbased graph decomposition of Räcke that was developed in the context of oblivious routing schemes, as well as, by the construction of the ultrasparsifiers due to Spielman and Teng that was employed to preconditioning symmetric diagonallydominant matrices. 1
Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey
, 2013
"... ..."
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...
C.: Higherorder segmentation via multicuts
 CoRR abs/1305.6387
"... Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, based on local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higherorder models provide a prominent class of representatives, ..."
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Cited by 6 (1 self)
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Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, based on local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higherorder models provide a prominent class of representatives, that cover a broad range of segmentation problems relevant to image analysis and computer vision. We show how to take into account such higherorder terms systematically in view of computational inference, and present results of a comprehensive and competitive numerical evaluation of a variety of dedicated cuttingplane algorithms. Our results reveal ways to evaluate a significant subset of models globally optimal, without compromising runtime. Polynomially solvable relaxations are studied as well, along with advanced rounding schemes for postprocessing.