NDLI: Approximation Algorithms for the 0-Extension Problem

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Calinescu, Gruia Rabani, Yuval Karloff, Howard
Copyright Year	2004
Abstract	In the 0-extension 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 et al. [SIAM J. Comput.}, 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23].We present approximation algorithms for {\sc 0-Extension}. 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 [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput.}, 25 (1996), pp. 235--251] 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 $c \sqrt{\lg k}$ 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.
Starting Page	358
Ending Page	372
Page Count	15
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539701395978
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	34
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-27
Access Restriction	Subscribed
Subject Keyword	metric space approximation algorithm linear programming relaxation Approximation algorithms graph partitioning
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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