On the complexity of schedule control problems for knockout tournaments

摘要

Knockout tournaments constitute a common format of sporting events, and also model a specific type of election scheme (namely, sequential pairwise elimination election). In such tournaments the designer controls the shape of the tournament (a binary tree) and the seeding of the players (their assignment to the tree leaves). In this paper we investigate the computational complexity of tournament schedule control, i.e., designing a tournament that maximizes the winning probability a target player. We start with a generic probabilistic model consisting of a matrix of pairwise winning probabilities, and then investigate the problem under two types of constraint: constraints on the probability matrix, and constraints on the allowable tournament structure. While the complexity of the general problem is as yet unknown, these various constraints -- all naturally occurring in practice -- serve to push to the problem to one side or the other: easy (polynomial) or hard (NP-complete).