Free access
Proceedings
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

A subexponential lower bound for the Random Facet algorithm for Parity Games

Abstract

Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of turn-based Stochastic Mean Payoff Games. It is a major open problem whether these game families can be solved in polynomial time.
The currently theoretically fastest algorithms for the solution of all these games are adaptations of the randomized algorithms of Kalai and of Matousek, Sharir and Welzl for LP-type problems, an abstract generalization of linear programming. The expected running time of both algorithms is subexponential in the size of the game, i.e., , where n is the number of vertices in the game. We focus in this paper on the algorithm of Matousek, Sharir and Welzl and refer to it as the Random Facet algorithm. Matoušek constructed a family of abstract optimization problems such that the expected running time of the Random Facet algorithm, when run on a random instance from this family, is close to the subexponential upper bound given above. This shows that in the abstract setting, the upper bound on the complexity of the Random Facet algorithm is essentially tight.
It is not known, however, whether the abstract optimization problems constructed by Matoušek correspond to games of any of the families mentioned above. There was some hope, therefore, that the Random Facet algorithm, when applied to, say, parity games, may run in polynomial time. We show, that this, unfortunately, is not the case by constructing explicit parity games on which the expected running time of the Random Facet algorithm is close to the subexponential upper bound. The games we use mimic the behavior of a randomized counter. They are also the first explicit LP-type problems on which the Random Facet algorithm is not polynomial.

Formats available

You can view the full content in the following formats:

Information & Authors

Information

Published In

cover image Proceedings
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 202 - 216
Editor: Dana Randall, Georgia Institute of Technology, Atlanta, Georgia
ISBN (Print): 978-0-898719-93-2
ISBN (Online): 978-1-61197-308-2

History

Published online: 18 December 2013

Authors

Affiliations

Notes

Supported by the Center for Algorithmic Game Theory, funded by the Carlsberg Foundation.
Research supported by ISF grant no. 1306/08.

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.