The discrete strategy improvement algorithm for parity games and complexity measures for directed graphs
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 superpolynomially many iteration steps, for all popular local improvements rules, including switchall (also with Fearnley's snare memorisation), switchbest, randomfacet, randomedge, switchhalf, leastrecentlyconsidered, and Zadeh's Pivoting rule. We analyse the examples provided by Friedmann in terms of complexity measures for directed graphs such as treewidth, DAGwidth, Kellywidth, 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 superpolynomial running time in the general case, where the problem of polynomialtime solvability is open, it even has superpolynomial lower time bounds on natural classes of parity games on which efficient algorithms are known.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.2459
 Bibcode:
 2012arXiv1210.2459C
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Computer Science and Game Theory
 EPrint:
 In Proceedings GandALF 2012, arXiv:1210.2028