TY - CHAP

T1 - The multi-multiway cut problem

AU - Avidor, Adi

AU - Langberg, Michael

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2004

Y1 - 2004

N2 - In this paper, we define and study a natural generalization of the multicut and multiway cut problems: the minimum multi-multiway cut problem. The input to the problem is a weighted undirected graph G = (V, E) and k sets S1,S2, ..., Sk of vertices. The goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets S1,S2, ..., Sk, i.e., disconnects every pair of vertices u and u such that u, v ∈ Si, for some i. This problem generalizes both the multicut problem, when |Si| = 2, for 1 ≤ i ≤ k, and the multiway cut problem, when k = 1. We present an approximation algorithm for the multi-multiway cut problem with an approximation ratio which matches that obtained by Garg, Vazirani, and Yannakakis [GVY96] on the standard multicut problem. Namely, our algorithm has an O(log 2k) approximation ratio. Moreover, we consider instances of the minimum multi-multiway cut problem which are known to have an optimal solution of light weight. We show that our algorithm has an approximation ratio substantially better than O(log 2k) when restricted to such "light" instances. Specifically, we obtain an O(log LP)-approximation algorithm for the problem, when all edge weights are at least 1, where LP is the value of a natural LP-relaxation of the problem. The latter improves the O (log LP log log LP) approximation ratio for the minimum multicut problem (implied by the work of Seymour [Sey95] and Even et al. [ENSS98]).

AB - In this paper, we define and study a natural generalization of the multicut and multiway cut problems: the minimum multi-multiway cut problem. The input to the problem is a weighted undirected graph G = (V, E) and k sets S1,S2, ..., Sk of vertices. The goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets S1,S2, ..., Sk, i.e., disconnects every pair of vertices u and u such that u, v ∈ Si, for some i. This problem generalizes both the multicut problem, when |Si| = 2, for 1 ≤ i ≤ k, and the multiway cut problem, when k = 1. We present an approximation algorithm for the multi-multiway cut problem with an approximation ratio which matches that obtained by Garg, Vazirani, and Yannakakis [GVY96] on the standard multicut problem. Namely, our algorithm has an O(log 2k) approximation ratio. Moreover, we consider instances of the minimum multi-multiway cut problem which are known to have an optimal solution of light weight. We show that our algorithm has an approximation ratio substantially better than O(log 2k) when restricted to such "light" instances. Specifically, we obtain an O(log LP)-approximation algorithm for the problem, when all edge weights are at least 1, where LP is the value of a natural LP-relaxation of the problem. The latter improves the O (log LP log log LP) approximation ratio for the minimum multicut problem (implied by the work of Seymour [Sey95] and Even et al. [ENSS98]).

UR - http://www.scopus.com/inward/record.url?scp=35048871602&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-27810-8_24

DO - 10.1007/978-3-540-27810-8_24

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AN - SCOPUS:35048871602

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 273

EP - 284

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - Hagerup, Torben

A2 - Katajainen, Jyrki

PB - Springer Verlag

ER -