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Tractable Unordered 3-CNF Games

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12118)

Abstract

The classic TQBF problem can be viewed as a game in which two players alternate turns assigning truth values to a CNF formula’s variables in a prescribed order, and the winner is determined by whether the CNF gets satisfied. The complexity of deciding which player has a winning strategy in this game is well-understood: it is \(\textsf {NL}\)-complete for 2-CNFs and \(\textsf {PSPACE}\)-complete for 3-CNFs.

We continue the study of the unordered variant of this game, in which each turn consists of picking any remaining variable and assigning it a truth value. The complexity of deciding who can win on a given CNF is less well-understood; prior work by the authors showed it is in \(\textsf {L}\) for 2-CNFs and \(\textsf {PSPACE}\)-complete for 5-CNFs. We conjecture it may be efficiently solvable on 3-CNFs, and we make progress in this direction by proving the problem is in \(\textsf {P}\), indeed in \(\textsf {L}\), for 3-CNFs with a certain restriction, namely that each width-3 clause has at least one variable that appears in no other clause. Another (incomparable) restriction of this problem was previously shown to be tractable by Kutz.

Keywords

  • 3-CNF
  • Games
  • Unordered
  • Logarithmic space

This work was supported by NSF grant CCF-1657377.

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References

  1. Ahlroth, L., Orponen, P.: Unordered constraint satisfaction games. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 64–75. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32589-2_9

    CrossRef  Google Scholar 

  2. Aspvall, B., Plass, M., Tarjan, R.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979)

    MathSciNet  CrossRef  Google Scholar 

  3. Byskov, J.: Maker-maker and maker-breaker games are PSPACE-complete. Technical report RS-04-14, BRICS, Department of Computer Science, Aarhus University (2004)

    Google Scholar 

  4. Calabro, C.: 2-TQBF is in P (2008). https://cseweb.ucsd.edu/~ccalabro/essays/complexity_of_2tqbf.pdf. Unpublished

  5. Kutz, M.: The angel problem, positional games, and digraph roots. Ph.D. thesis, Freie Universität Berlin (2004). Chapter 2: Weak Positional Games

    Google Scholar 

  6. Kutz, M.: Weak positional games on hypergraphs of rank three. In: Proceedings of the 3rd European Conference on Combinatorics, Graph Theory, and Applications (EuroComb), pp. 31–36. Discrete Mathematics & Theoretical Computer Science (2005)

    Google Scholar 

  7. Rahman, M.L., Watson, T.: Complexity of unordered CNF games. In: Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC), pp. 9:1–9:12. Schloss Dagstuhl (2018)

    Google Scholar 

  8. Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17:1–17:24 (2008)

    Google Scholar 

  9. Rozenman, E., Vadhan, S.: Derandomized squaring of graphs. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX/RANDOM -2005. LNCS, vol. 3624, pp. 436–447. Springer, Heidelberg (2005). https://doi.org/10.1007/11538462_37

    CrossRef  Google Scholar 

  10. Schaefer, T.: Complexity of decision problems based on finite two-person perfect-information games. In: Proceedings of the 8th Symposium on Theory of Computing (STOC), pp. 41–49. ACM (1976)

    Google Scholar 

  11. Schaefer, T.: On the complexity of some two-person perfect-information games. J. Comput. Syst. Sci. 16(2), 185–225 (1978)

    MathSciNet  CrossRef  Google Scholar 

  12. Stockmeyer, L., Meyer, A.: Word problems requiring exponential time. In: Proceedings of the 5th Symposium on Theory of Computing (STOC), pp. 1–9. ACM (1973)

    Google Scholar 

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Correspondence to Md Lutfar Rahman .

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Rahman, M.L., Watson, T. (2020). Tractable Unordered 3-CNF Games. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_29

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  • DOI: https://doi.org/10.1007/978-3-030-61792-9_29

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