Tournament Fixing Parameterized by Feedback Vertex Set Number Is FPT

Abstract

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 as.

How to Make Knockout Tournaments More Popular?

“…To show Theorem 15, we adapt an algorithm for TOUR-NAMENT FIXING parameterized by the feedback vertex number by Zehavi (2023). Due to the similarity of the algorithms, we need to introduce many concepts of Zehavi (2023).…”

“…To show Theorem 15, we adapt an algorithm for TOUR-NAMENT FIXING parameterized by the feedback vertex number by Zehavi (2023). Due to the similarity of the algorithms, we need to introduce many concepts of Zehavi (2023). In TOURNAMENT FIXING, we are asked whether a seeding exists that makes a specific player win the tournament, given a set of players and a tournament graph that defines the outcome of a game between each pair of players.…”

“…In particular, when the aforementioned information is deterministic and complete (i.e., for every possible match involving two players, we "know" who will be the winner) and our only influence on the competition is by picking the seeding, the problem is known as the TOURNAMENT FIXING problem. This problem and its various variations have been extensively studied from the viewpoints of classical complexity, parameterized complexity, and structural analysis (see, e.g., (Gupta et al 2019(Gupta et al , 2018aKim and Williams 2015;Aziz et al 2018;Vu, Altman, and Shoham 2009;Williams 2010;Zehavi 2023;Ramanujan and Szeider 2017;Stanton and Williams 2011b,a;Kim, Suksompong, and Williams 2017;Manurangsi and Suksompong 2023); this list is illustrative rather than comprehensive).…”

“…We design a fixedparameter algorithm for this parameter as well. To this end, we adapt the recent algorithm of Zehavi (2023) for TOUR-NAMENT FIXING parameterized by the feedback vertex set number of the prediction graph of the input, being the complete digraph obtained by having an arc from a player a to a player b if a is predicted to beat b. The motivation to consider this parameter is that, quite often, only a small set of players are truly profitable or popular.…”

Threshold-Based Responsive Simulated Annealing for Directed Feedback Vertex Set Problem

“…It involves obtaining a directed acyclic graph by removing as few vertices as possible from a given directed graph. DFVSP has wide applications in various domains, ranging from Very Large Scale Integration (VLSI) circuit design (Hudli and Hudli 1994;Orenstein, Kohavi, and Pomeranz 1995), partial scan design of circuit (Lee and Reddy 1990), program verification (Seymour 1995), deadlock resolution (Jain, Hajiaghayi, and Talwar 2005), network attack (Mugisha and Zhou 2016), constraint satisfaction (Bar-Yehuda et al 1994), Bayesian inference (Bar-Yehuda et al 1998), tournament (Ramanujan andSzeider 2017;Zehavi 2023), and so on.…”