A Polynomial Kernel for Multicut in Trees

1 ENS Ca han, 61, avenue du Président Wilson, 94235 Ca han edex Fran e E-mail address: nbousque dptinfo.ensa han.fr 2 Université Montpellier II CNRS, LIRMM, 161 rue Ada 34392 Montpellier Cedex 5 Fran e E-mail address: daligault lirmm.fr E-mail address: thomasse lirmm.fr 3 Royal Holloway, University of London, Egham Hill, EGHAM, TW20 0EX UK E-mail address: anders s.rhul.a .uk Abstra t. The MULTICUT IN TREES problem onsists in de iding, given a tree, a set of requests (i.e. paths in the tree) and an integer k, whether there exists a set of k edges utting all the requests. This problem was shown to be FPT by Guo and Niedermeyer in [10℄. They also provided an exponential kernel. They asked whether this problem has a polynomial kernel. This question was also raised by Fellows in [1℄. We show that MULTICUT IN TREES has a polynomial kernel. 1. Introdu tion An e ient way of dealing with NP-hard problems is to identify a parameter whi h ontains its omputational hardness. For instan e, instead of asking for a minimum vertex over in a graph a lassi al NP-hard optimization question one an ask for an algorithm whi h would de ide, in O(f(k).n) time for some xed d, if a graph of size n has a vertex over of size at most k. If su h an algorithm exists, the problem is alled xed-parameter tra table, or FPT for short. An extensive litterature is devoted to FPT, the reader is invited to read [4℄, [7℄ and [12℄. Kernelization is a natural way of proving that a problem is FPT. Formally, a kernelization algorithm re eives as input an instan e (I, k) of the parameterized problem, and outputs, in polynomial time in the size of the instan e, another instan e (I , k) su h that • k ≤ k, • the size of I ′ only depends of k, • the instan es (I, k) and (I , k) are both true or both false. Part of this resear h was supported by Allian e Proje t "Partitions de graphes orientés". Part of this resear h was supported by ANR Proje t GRAAL.