Constant ratio fixed-parameter approximation of the edge multicut problem

The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s ₁,t ₁}, ..., {s _m ,t _m }; the task is to remove a minimum set of edges such that s _i and t _i are disconnected for every 1≤i≤m. The parameterized complexity of the problem, parameterized by the maximum number k of edges that are allowed to be removed, is currently open. The main result of the paper is a parameterized 2-approximation algorithm: in time f(k)•n ^O(1), we can either find a solution of size 2k or correctly conclude that there is no solution of size k. The proposed algorithm is based on a transformation of the Edge Multicut problem into a variant of parameterized Max-2-SAT problem, where the parameter is related to the number of clauses that are not satisfied. It follows from previous results that the latter problem can be 2-approximated in a fixed-parameter time; on the other hand, we show here that it is W[1]-hard. Thus the additional contribution of the present paper is introducing the first natural W[1]-hard problem that is constant-ratio fixed-parameter approximable.