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

The problem whether winning regions and wining strategies for parity games can be computed in polynomial time is a major open problem in the field of infinite games, which is relevant for many applications in logic and formal verification. For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been considered to be a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that this 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 time lower bounds on natural classes of parity games on which efficient algorithms are known.