Approximability and parameterized complexity of minmax values

TL;DR

This paper studies the complexity of approximating the minmax value in multi-player games, establishing NP-hardness for high precision, quasi-polynomial algorithms for moderate precision, and parameterized complexity results including fixed-parameter tractability and hardness.

Contribution

It provides tight bounds on the approximability of minmax values, analyzes the problem's parameterized complexity, and identifies cases where exact computation is efficient versus cases that are W[1]-hard.

Findings

High-precision approximation is NP-hard.

Moderate-precision approximation is quasi-polynomial time.

Exact computation is polynomial time for certain restricted cases.

Abstract

We consider approximating the minmax value of a multi-player game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of epsilon log n digits (for any constant epsilon>0 is NP-hard, where n is the size of the game. On the other hand, approximating the value with a precision of c log log n digits (for any constant c >= 1) can be done in quasi-polynomial time. We consider the parameterized complexity of the problem, with the parameter being the number of pure strategies k of the player for which the minmax value is computed. We show that if there are three players, k=2 and there are only two possible rational payoffs, the minmax value is a rational number and can be computed exactly in linear time. In the general case, we show that the value can be approximated with any polynomial number of digits of accuracy in time n^(O(k)). On the other hand, we show that minmax value approximation is W[1]-hard and hence not likely to be fixed parameter tractable. Concretely, we show that if k-CLIQUE requires time n^(Omega(k)) then so does minmax value computation.