{"id":157278,"date":"2006-01-01T00:00:00","date_gmt":"2006-01-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/approximating-the-k-multicut-problem\/"},"modified":"2018-10-16T21:50:58","modified_gmt":"2018-10-17T04:50:58","slug":"approximating-the-k-multicut-problem","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/approximating-the-k-multicut-problem\/","title":{"rendered":"Approximating the k-multicut problem"},"content":{"rendered":"<p>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> \u2264 <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 + \u03b5, for any fixed \u03b5 > 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 \u03b5 > 0, we also obtain a polynomial time algorithm for <i>k<\/i>-multicut on trees which returns a solution of cost at most (2 + \u03b5) \u00b7 <i>OPT<\/i>, that separates at least (1 &#8211; \u03b5) \u00b7 <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].<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study the k-multicut problem: Given an edge-weighted undirected graph, a set of l pairs of vertices, and a target k \u2264 l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the k-multicut problem on [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"mohsingh"}],"msr_publishername":"ACM 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