A Polynomial Kernel for Feedback Arc Set on Bipartite Tournaments

Abstract

In the k -Feedback Arc/Vertex Set problem we are given a directed graph D and a positive integer k and the objective is to check whether it is possible to delete at most k arcs/vertices from D to make it acyclic. Dom et al.[CIAC, 2006] initiated a study of the Feedback Arc Set problem on bipartite tournaments (k -FASBT) in the realm of parameterized complexity. They showed that k -FASBT can be solved in time O(3.373 k n 6) on bipartite tournaments having n vertices. However, until now there was no known polynomial sized problem kernel for k -FASBT. In this paper we obtain a cubic vertex kernel for k -FASBT. This completes the kernelization picture for the Feedback Arc/Vertex Set problem on tournaments and bipartite tournaments, as for all other problems polynomial kernels were known before. We obtain our kernel using a non-trivial application of “independent modules” which could be of independent interest.