Combinatorial simplex algorithms can solve mean payoff games
Abstract
A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games. Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.
Cite
@article{arxiv.1309.5925,
title = {Combinatorial simplex algorithms can solve mean payoff games},
author = {Xavier Allamigeon and Pascal Benchimol and Stéphane Gaubert and Michael Joswig},
journal= {arXiv preprint arXiv:1309.5925},
year = {2015}
}
Comments
v1: 15 pages, 3 figures; v2: improved presentation, introduction expanded, 18 pages, 3 figures