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The Parameterized Complexity of Positional Games

Authors Édouard Bonnet, Serge Gaspers, Antonin Lambilliotte, Stefan Rümmele, Abdallah Saffidine

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Keywords
  • Hex
  • Maker-Maker games
  • Maker-Breaker games
  • Enforcer-Avoider games
  • parameterized complexity theory

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Abstract

We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows’ influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is W[1]-complete when parameterized by formula size. We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices. In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise. Short Maker-Maker, Short Maker-Breaker, and Short Enforcer-Avoider are respectively AW[*]-, W[1]-, and co-W[1]-complete parameterized by the number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first level of the W-hierarchy when the winning condition only depends on which vertices one player has been able to pick, but AW[*]-complete when it depends on which vertices both players have picked. However, some positional games with highly structured board and winning configurations are fixed-parameter tractable. We give another example of such a game, Short k-Connect, which is fixed-parameter tractable when parameterized by the number of moves.

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Édouard Bonnet, Serge Gaspers, Antonin Lambilliotte, Stefan Rümmele, and Abdallah Saffidine. The Parameterized Complexity of Positional Games. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 90:1-90:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ICALP.2017.90

Author Details

Édouard Bonnet
Serge Gaspers
Antonin Lambilliotte
Stefan Rümmele
Abdallah Saffidine

References

  1. Karl R. Abrahamson, Rodney G. Downey, and Michael R. Fellows. Fixed-parameter intractability II (extended abstract). In Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science (STACS 93), volume 665 of Lecture Notes in Computer Science, pages 374-385. Springer, 1993. URL: http://dx.doi.org/10.1007/3-540-56503-5_38.
  2. Karl R. Abrahamson, Rodney G. Downey, and Michael R. Fellows. Fixed-parameter tractability and completeness IV: on completeness for W[P] and PSPACE analogues. Annals of Pure and Applied Logic, 73(3):235-276, 1995. URL: http://dx.doi.org/10.1016/0168-0072(94)00034-Z.
  3. Édouard Bonnet, Serge Gaspers, Antonin Lambilliotte, Stefan Rümmele, and Abdallah Saffidine. The parameterized complexity of positional games. CoRR, abs/1704.08536, 2017. URL: https://arxiv.org/abs/1704.08536.
  4. Édouard Bonnet, Florian Jamain, and Abdallah Saffidine. On the complexity of connection games. Theoretical Computer Science, 644:2-28, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2016.06.033.
  5. Jesper Makholm Byskov. Maker-Maker and Maker-Breaker games are PSPACE-complete. Technical Report RS-04-14, BRICS, Department of Computer Science, Aarhus University, 2004. Google Scholar
  6. Rodney G. Downey and Michael R. Fellows. Parameterized complexity. Springer-Verlag, New York, 1999. Google Scholar
  7. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. Google Scholar
  8. Shimon Even and Robert Endre Tarjan. A combinatorial problem which is complete in polynomial space. Journal of the ACM, 23(4):710-719, 1976. URL: http://dx.doi.org/10.1145/321978.321989.
  9. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 1998. Google Scholar
  10. Fedor V. Fomin and Dániel Marx. FPT suspects and tough customers: Open problems of downey and fellows. In Hans L. Bodlaender, Rod Downey, Fedor V. Fomin, and Dániel Marx, editors, The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture Notes in Computer Science, pages 457-468. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-30891-8_19.
  11. Markus Frick and Martin Grohe. Deciding first-order properties of locally tree-decomposable structures. Journal of the ACM, 48(6):1184-1206, 2001. Google Scholar
  12. Martin Grohe and Stefan Wöhrle. An existential locality theorem. Annals of Pure and Applied Logic, 129(1):131-148, 2004. URL: http://dx.doi.org/10.1016/j.apal.2004.01.005.
  13. Alfred W. Hales and Robert I. Jewett. Regularity and positional games. Transactions of the American Mathematical Society, 106:222-229, 1963. Google Scholar
  14. Ming Yu Hsieh and Shi-Chun Tsai. On the fairness and complexity of generalized k-in-a-row games. Theoretical Computer Science, 385(1):88-100, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.05.031.
  15. Stefan Reisch. Gobang ist PSPACE-vollständig. Acta Informatica, 13(1):59-66, 1980. Google Scholar
  16. Stefan Reisch. Hex ist PSPACE-vollständig. Acta Informatica, 15:167-191, 1981. URL: http://dx.doi.org/10.1007/BF00288964.
  17. Thomas J. Schaefer. On the complexity of some two-person perfect-information games. Journal of Computer and System Sciences, 16(2):185-225, 1978. URL: http://dx.doi.org/10.1016/0022-0000(78)90045-4.
  18. Allan Scott. On the Parameterized Complexity of Finding Short Winning Strategies in Combinatorial Games. PhD thesis, University of Victoria, 2009. Google Scholar
  19. Allan Scott and Ulrike Stege. Parameterized chess. In Proceedings of the 3rd International Workshop on Parameterized and Exact Computation (IWPEC 2008), volume 5018 of Lecture Notes in Computer Science, pages 172-189. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-79723-4_17.
  20. Allan Scott and Ulrike Stege. Parameterized pursuit-evasion games. Theoretical Computer Science, 411(43):3845-3858, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.07.004.
  21. Detlef Seese. Linear time computable problems and first-order descriptions. Mathematical Structures in Computer Science, 6(6):505-526, 1996. Google Scholar
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