Topics in fixing knockout tournaments : bribery, NP-hardness, and parameterization

Abstract

In this thesis, we analyze the problem of fixing balanced knockout tournaments by arranging the tournament's initial seeding to guarantee one player's victory. We characterize the problem's hardness through a variety of perspectives. First, we investigate the computational complexity of fixing the tournament while bribing players to lose matches they would typically win. We give a model of bribery in which one is given a pairwise comparison matrix whose entries contain the probability of the row player beating another column player in a match, where the organizer's ability to bribe players is constrained by the cost of each bribe and a fixed budget, and where the tournament seeding can be manipulated arbitrarily. We show that it is NP- hard to find a bribery scheme and a seeding under which a given player always wins the tournament, even when the original pairwise comparison matrix is monotonic; the hardness of fixing a tournament in this case without bribery is open. We also show that when the probability matrix is binary, for almost all n-player inputs generated by the Condorcet random model, if one bribes a specific number of the "top" O(log n) players, then there is an efficiently constructible winning seeding for any player. Next, we investigate the relationship of the deterministic case of the tournament fixing problem to other NP-complete problems. We demonstrate the futility of constructing a reduction to it from certain well-known graph problems, showing why the features of these problems are ultimately incompatible. We analyze the blowup of the NP-hardness reduction to the problem from a restricted version of 3SAT and use it to give a direct NP-hardness reduction from 3SAT. Finally, we apply parameterized complexity to the deterministic tournament fixing problem, giving a simpler algorithm that matches the runtime of the fastest known algorithm, using the size of the input tournament graph's feedback arc set as a parameter.