Single parameter FPT-algorithms for non-trivial games

Abstract

We know that k-Uniform Nash is W[2]-Complete when we consider imitation symmetric win-lose games (with k as the parameter) even when we have two players. However, this paper provides positive results regarding Nash equilibria. We show that consideration of sparse games or limitations of the support result in fixed-parameter algorithms with respect to one parameter only for the k-Uniform Nash problem. That is, we show that a sample uniform Nash equilibrium in r-sparse imitation symmetric win-lose games is not as hard because it can be found in FPT time (i.e polynomial in the size of the game, but maybe exponential in r). Moreover, we show that, although NP-Complete, the problem of Best Nash Equilibrium is also x-parameter tractable.