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Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

David Karger·Phil Klein·Cliff Stein·Mikkel Thorup·Neal E. Young
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Abstract

The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on the observation that the problem of designing a randomized rounding scheme for a geometric relaxation is itself a linear programming problem. The paper explores computational solutions to this problem, and gives a proof that for a general class of geometric relaxations, there are always randomized rounding schemes that match the integrality gap.

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David Karger, Phil Klein, Cliff Stein, Mikkel Thorup, Neal E. Young. 2002-05-19. Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut. https://doi.org/10.1287/moor.1030.0086

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