Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T\S is acyclic. In this article we consider the FEEDBACK VERTEX SET problem in tournaments. Here the input is a tournament T and an integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for FEEDBACK VERTEX SET IN TOURNAMENTS. The running time of our algorithm is upper-bounded by O(1.6181^k + n^{O(1)}) and by O(1.466^n). Thus our algorithm simultaneously improves over the fastest known parameterized algorithm for the problem by Dom et al. running in time O(2^kk^{O(1)} + n^{O(1)}), and the fastest known exact exponential-time algorithm by Gaspers and Mnich with running time O(1.674^n). On the way to proving our main result we prove a strengthening of a special case of a graph partitioning theorem due to Bollobas and Scott. In particular we show that the vertices of any undirected m-edge graph of maximum degree d can be colored white or black in such a way that for each of the two colors, the number of edges with both endpoints of that color is between m/4-d/2 and m/4+d/2.