Approximation algorithms for requirement cut on graphs

Authors:
Viswanath Nagarajan;Ramamoorthi Ravi
Affiliations:
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA;Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA
Venue:
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Year:
2005

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Abstract

In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to a requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X1,⋯,Xg⊆V, each with a requirement ri between 0 and |Xi|. The goal is to find a minimum cost set of edges whose removal separates each group Xi into at least ri disconnected components. We give an O(log n log (gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log n log (gR)). On trees, we obtain an improved guarantee of O(log (gR)). There is a natural Ω (log g) hardness of approximation for the requirement cut problem.