Kernels for Feedback Arc Set In Tournaments

A tournament T = (V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed > 0, the kernelized instance has at most (2 + )k vertices. Our result improves the previous known bound of O(k) on the kernel size for k-FAST. For our kernelization algorithm we find a subclass of tournaments where one can find a minimum sized feedback arc set in polynomial time and use the known polynomial time approximation scheme for k-FAST.