The discrete strategy improvement algorithm for parity games and complexity measures for directed graphs

TL;DR

This paper analyzes the limitations of the discrete strategy improvement algorithm for parity games, showing it requires super-polynomial time even on classes of graphs where efficient algorithms exist, thus highlighting its significant computational constraints.

Contribution

The paper provides a detailed complexity measure analysis of Friedmann's counterexamples, demonstrating the algorithm's limitations across various graph parameters.

Findings

The algorithm has super-polynomial lower bounds on many graph classes.

Friedmann's examples are bounded in complexity measures like treewidth and DAG-width.

The analysis reveals the algorithm's limitations are more extensive than previously thought.

Abstract

For some time the discrete strategy improvement algorithm due to Jurdzinski and Voge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule. We analyse the examples provided by Friedmann in terms of complexity measures for directed graphs such as treewidth, DAG-width, Kelly-width, entanglement, directed pathwidth, and cliquewidth. It is known that for every class of parity games on which one of these parameters is bounded, the winning regions can be efficiently computed. It turns out that with respect to almost all of these measures, the complexity of Friedmann's counterexamples is bounded, and indeed in most cases by very small numbers. This analysis strengthens in some sense Friedmann's results and shows that the discrete strategy improvement algorithm is even more limited than one might have thought. Not only does it require super-polynomial running time in the general case, where the problem of polynomial-time solvability is open, it even has super-polynomial lower time bounds on natural classes of parity games on which efficient algorithms are known.