Abstract
Determining the winner of a Parity Game is a major problem in computational complexity with a number of applications in verification. In a parameterized complexity setting, the problem has often been considered with parameters such as (directed versions of) treewidth or clique-width, by applying dynamic programming techniques.
In this paper we adopt a parameterized approach which is more inspired by well-known (non-parameterized) algorithms for this problem. We consider a number of natural parameterizations, such as by Directed Feedback Vertex Set, Distance to Tournament, and Modular Width. We show that, for these parameters, it is possible to obtain recursive parameterized algorithms which are simpler, faster and only require polynomial space. We complement these results with some algorithmic lower bounds which, among others, rule out a possible avenue for improving the best-known sub-exponential time algorithm for parity games.
J. Gajarský—Supported by the research centre Institute for Theoretical Computer Science (CE-ITI), project P202/12/G061.
V. Mitsou—Supported by ERC Starting Grant PARAMTIGHT (No. 280152).
S. Ordyniak—Supported by Employment of Newly Graduated Doctors of Science for Scientific Excellence (CZ.1.07/2.3.00/30.0009).
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Notes
- 1.
As usual in parameterized complexity the \(O^*\!\!\left( f(k)\right) \)-notation, for some function f of the parameter k, means that there is an algorithm running in time \(O(f(k)n^{O(1)})\), where n is the input size of the problem.
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Acknowledgements
We would like to thank Danupon Nanongkai for suggesting this problem and for our useful discussions.
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Gajarský, J., Lampis, M., Makino, K., Mitsou, V., Ordyniak, S. (2015). Parameterized Algorithms for Parity Games. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_28
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