Winning a tournament by any means necessary

In a tournament, n players enter the competition. In each round, they are paired-up to compete against each other. Losers are thrown, while winners proceed to the next round, until only one player (the winner) is left. Given a prediction of the outcome, for every pair of players, of a match between them (modeled by a digraph D), the competitive nature of a tournament makes it attractive for manipulators. In the Tournament Fixing (TF) problem, the goal is to decide if we can conduct the competition (by controlling how players are paired-up) so that our favorite player w wins. A common form of manipulation is to bribe players to alter the outcome of matches. Kim and Williams [IJCAI 2015] integrated such deceit into TF, and showed that the resulting problem is NP-hard when ` < (1 − ) log n alterations are possible (for any fixed > 0). For this problem, our contribution is fourfold. First, we present two operations that “obfuscate deceit”: given one solution, they produce another solution. Second, we present a combinatorial result, stating that there is always a solution with all reversals incident to w and “elite players”. Third, we give a closed formula for the case where D is a DAG. Finally, we present exact exponential-time and parameterized algorithms for the general case.