Abstract
Parity games are a much researched class of games in NP ∩ CoNP that are not known to be in P. Consequently, researchers have considered specialised algorithms for the case where certain graph parameters are small. In this paper, we show that, if a tree decomposition is provided, then parity games with bounded treewidth can be solved in O(k 3k + 2 ·n 2 ·(d + 1)3k) time, where n, k, and d are the size, treewidth, and number of priorities in the parity game. This significantly improves over previously best algorithm, given by Obdržálek, which runs in \(O(n \cdot d^{2(k+1)^2})\) time. Our techniques can also be adapted to show that the problem lies in the complexity class NC2, which is the class of problems that can be efficiently parallelized. This is in stark contrast to the general parity game problem, which is known to be P-hard, and thus unlikely to be contained in NC.
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References
Allender, E., Loui, M.C., Regan, K.W.: Complexity classes. In: Atallah, M.J., Blanton, M. (eds.) Algorithms and Theory of Computation Handbook, p. 22. Chapman & Hall/CRC (2010)
Amir, E.: Efficient approximation for triangulation of minimum treewidth. In: Proc. of UAI, pp. 7–15. Morgan Kaufmann Publishers Inc., San Francisco (2001)
Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-Width and Parity Games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)
Berwanger, D., Grädel, E.: Entanglement – A Measure for the Complexity of Directed Graphs with Applications to Logic and Games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25(6), 1305–1317 (1996)
Emerson, E.A., Jutla, C.S.: Tree automata, μ-calculus and determinacy. In: Proc. of FOCS, pp. 368–377. IEEE Computer Society Press (October 1991)
Emerson, E.A., Jutla, C.S., Sistla, A.P.: On Model-Checking for Fragments of μ-Calculus. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 385–396. Springer, Heidelberg (1993)
Fearnley, J., Lachish, O.: Parity Games on Graphs with Medium Tree-Width. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 303–314. Springer, Heidelberg (2011)
McNaughton, R.: Infinite games played on finite graphs. Ann. Pure Appl. Logic 65(2), 149–184 (1993)
Mostowski, A.W.: Games with forbidden positions. Technical Report 78, University of Gdańsk (1991)
Obdržálek, J.: Fast Mu-Calculus Model Checking when Tree-Width Is Bounded. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 80–92. Springer, Heidelberg (2003)
Stirling, C.: Local Model Checking Games (Extended Abstract). In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)
Thorup, M.: All structured programs have small tree width and good register allocation. Information and Computation 142(2), 159–181 (1998)
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Fearnley, J., Schewe, S. (2012). Time and Parallelizability Results for Parity Games with Bounded Treewidth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31585-5_20
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DOI: https://doi.org/10.1007/978-3-642-31585-5_20
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