Fixed‐parameter tractability and data reduction for multicut in trees

Abstract

We study an NP-complete (and MaxSNP-hard) communication problem on tree networks, the so-called Multicut in Trees: given an undirected tree and some pairs of nodes of the tree, find out whether there is a set of at most k tree edges whose removal separates all given pairs of nodes. Multicut has been intensively studied for trees as well as for general graphs mainly from the viewpoint of polynomialtime approximation algorithms. By way of contrast, we provide a

Two fixed-parameter algorithms for Vertex Covering by Paths on Trees

“…search tree and a size-O(k 2 ) problem kernel for unweighted VCPT, in case of Multicut in Trees we have a (simple) size-O(2 k ) search tree but only an exponential-size problem kernel [2]. This somehow opposite behavior in terms of parameterized complexity analysis of two fairly similar problems might stir interest in further research investigating both problems.…”

“…Compare the fixed-parameter algorithms for unweighted VCPT and the related Multicut in Trees problem [2] both with respect to the parameter solution size k. In the latter case, one is also given a tree together with a set of paths and the task is to find at most k edges that cut all the given paths.…”

A logical approach to multicut problems

“…While a number of results regarding the complexity of MULTICUT in several classes of graphs, such as interval graphs, trees, and series-parallel graphs have been shown to be FPT for some parameters [2,7,9], for general graphs a single parameter has not yet been isolated for which the problems are FPT.…”

“…Several forms of MULTICUT have been considered and have given rise to complexity studies [2,4,7,9,8,10]. A large part of the literature deals with the EDGE MUL-TICUT variant in which the solution is a set of edges E ⊆ E which, if removed, separates every terminal pair, but the variants in which sets of vertices are removed, UNRESTRICTED VERTEX MULTICUT, wherein any vertices can appear in the solution set V and RE-STRICTED VERTEX MULTICUT, wherein no terminal vertices can appear in the solution set, have also been studied [2].…”

Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

“…The major part of the literature deals with the "edge deletion variant" of Multicut (Edge Multicut) [8,10,16,21,20]. Given a graph and m pairs of terminal vertices, this problem is solvable in polynomial time for m = 1 [11] and m = 2 [22,23].…”