Tournament Fixing Parameterized by Feedback Vertex Set Number Is FPT

A knockout (or single-elimination) tournament is a format of a competition that is very popular in practice (particularly in sports, elections and decision making), and which has been extensively and intensively studied from a theoretical point of view for more than a decade. Particular attention has been devoted to the TOURNAMENT FIXING problem, where, roughly speaking, the objective is to determine whether we can conduct the knockout tournament in a way that makes our favorite player win. Here, part of the input is a tournament graph D that encodes the winner of each possible match. A sequence of papers has studied the parameterized complexity of TOURNAMENT FIXING with respect to the feedback arc set number (fas) of D. Given that this parameter yielded tractability, it has been asked explicitly and repeatedly whether TOURNAMENT FIXING is FPT also with respect to the feedback vertex set number (fvs) of D. We answer this question positively. In fact, although fvs can be arbitrarily smaller than fas, we attain the same dependency on the parameter in the time complexity. So, additionally, our work subsumes the best known algorithm for TOURNAMENT FIXING with respect to fas.