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Hex ist PSPACE-vollständig

Hex is PSPACE-complete


There are a number of board games such as Checkers [2], Go [5], and Gobang [8], which are known to be PSPACE-hard. This means that the problem to determine the player having a winning strategy in a given situation on an n×n board of one of these games is as hard to solve as everything computable in polynomial space. PSPACE-completeness has been previously proven for some combinatorial games played on graphs or by logical formulas [1, 9].

In this paper we will show that the same holds for the game of Hex. The crucial point of the proof is to establish PSPACE-hardness for a generalization of Hex played on planar graphs. This will be done by showing that the problem, whether a given quantified Boolean formula in conjunctive normal form is true, is polynomial time-reducible to the decision problem for generalized Hex. In order to do this we will use methods, which were also used to prove PSPACE-completeness of planar Geography in [5]. Therefore our proof is quite different from the proof provided by Even and Tarjan [1], who showed PSPACE-completeness of generalized Hex played on arbitrary graphs. Since it is easy to see that the decision problem for Hex is in PS PACE, the decision problem for Hex is PSPACE-complete.

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  1. Even, S., Tarjan, R.E.: A combinatorial problem which is complete in polynomial space. 7th annual ACM Symposium on Theory of Computing, 1975, pp 66–71

  2. Fraenkel, A.S., Garey, M.R., Johnson, D.S., Schaefer, T.J., Yesha, Y.: The complexity of checkers on an N×N board. 17th annual Symposium on Foundations of Computer Science. IEEE Computer Society 1978, pp 55–64

  3. Gardner, M.: Mathematische Rätsel und Probleme. Braunschweig: Vieweg, s. 33–39, 1964

    Google Scholar 

  4. Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. J. Assoc. Comput. Mach. 21, 549–568 (1974)

    Google Scholar 

  5. Lichtenstein, D., Sipser, M.: Go is PSPACE-hard. 19th annual Symposium on Foundations of Computer Science. IEEE Computer Society 1978, pp 48–54

  6. Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time. 5th annual ACM Symposium on Theory of Computing, 1973, pp 1–9

  7. Paul, W.J.: Komplexitätstheorie. Stuttgart: Teubner 1978

    Google Scholar 

  8. Reisch, S.: Gobang is PSPACE-vollständig. Acta Informat. 13, 59–66 (1980)

    Google Scholar 

  9. Schaefer, T.J.: On the complexity of some zwo-person perfect-information games. J. Comput. System Sci. 16, 185–225 (1978)

    Google Scholar 

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Reisch, S. Hex ist PSPACE-vollständig. Acta Informatica 15, 167–191 (1981).

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