Approximating the k-multicut problem

We study the <i>k</i>-multicut problem: Given an edge-weighted undirected graph, a set of <i>l</i> pairs of vertices, and a target <i>k</i> ≤ <i>l</i>, find the minimum cost set of edges whose removal disconnects <i>at least k</i> pairs. This generalizes the well known multicut problem, where <i>k = l.</i> We show that the <i>k</i>-multicut problem on trees can be approximated within a factor of 8/3 + ε, for any fixed ε > 0, and within <i>O</i>(log<sup>2</sup> <i>n</i> log log <i>n</i>) on general graphs, where <i>n</i> is the number of vertices in the graph.For any fixed ε > 0, we also obtain a polynomial time algorithm for <i>k</i>-multicut on trees which returns a solution of cost at most (2 + ε) · <i>OPT</i>, that separates at least (1 - ε) · <i>k</i> pairs, where <i>OPT</i> is the cost of the optimal solution separating <i>k</i> pairs.Our techniques also give a simple 2-approximation algorithm for the multicut problem on trees using total unimodularity, matching the best known algorithm [8].

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