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Multicut in trees viewed through the eyes of vertex cover.pdf

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Multicut in trees viewed through the eyes of vertex cover.pdf

Journal of Computer and System Sciences 78 (2012) 1637–1650 Contents lists available at SciVerse ScienceDirect Journal of Computer and System Sciences /locate/jcss Multicut in trees viewed through the eyes of vertex cover ? Jianer Chen a,1, Jia-Hao Fan a, Iyad Kanj b,?,2, Yang Liu c, Fenghui Zhang d a Department of Computer Science and Engineering, Texas AM University, College Station, TX 77843, USA b DePaul University, Chicago, IL 60604, USA c Department of Computer Science, University of Texas-Pan American, Edinburg, TX 78539, USA d Google Kirkland, 747 6th Street South, Kirkland, WA 98033, USA article info Article history: Received 9 September 2011 Received in revised form 23 February 2012 Accepted 2 March 2012 Available online 6 March 2012 Keywords: Parameterized complexity Kernelization Parameterized algorithms Multicut abstract We take a new look at the multicut problem in trees, denoted multicut on trees henceforth, through the eyes of the vertex cover problem. This connection, together with other techniques that we develop, allows us to give an upper bound of O (k3) on the kernel size for multicut on trees, signi?cantly improving the O (k6) upper bound given by Bousquet et al. We exploit this connection further for multicut on trees that runs in time O ?(ρk), to present a √parameterized algorithm where ρ = ( 5 + 1)/2 ≈ 1.618. This improves the previous (time) upper bound of O ?(2k), given by Guo and Niedermeier, for the problem. ? 2012 Elsevier Inc. All rights reserved. 1. Introduction In the multicut on trees problem we are given a tree T , a set of requests R ? V (T ) × V (T ) between pairs of vertices in T , and a nonnegative integer k, and we are asked to decide if we can remove at most k edges from the tree to disconnect all the requests in R (i.e., every path in the tree that corresponds to a request in R contains at least one of the removed edges). The multicut on trees problem has applications in networking [5]. The problem is known to be NP-hard, an