Abstract
A black-white combinatorial game is a two-person game in which the pieces are colored either black or white. The players alternate moving or taking elements of a specific color designated to them before the game begins. A player loses the game if there is no legal move available for his color on his turn.
We first show that some black-white versions of combinatorial games can only assume combinatorial game values that are numbers, which indicates that the game has many nice properties making it easier to solve. Indeed, numeric games have only previously been shown to be hard for \(\mathsf{NP}\). We exhibit a language of natural numeric games (specifically, black-white poset games) that is \(\mathsf{PSPACE}\)-complete, closing the gap in complexity for the first time between these numeric games and the large collection of combinatorial games that are known to be \(\mathsf{PSPACE}\)-complete.
In this vein, we also show that the game of Col played on general graphs is also \(\mathsf{PSPACE}\)-complete despite the fact that it can only assume two very simple game values. This is interesting because its natural black-white variant is numeric but only complete for \(\mathsf{P}^{\mathsf{NP}[\log ]}\). Finally, we show that the problem of determining the winner of black-white Graph Nim is in \(\mathsf{P}\) using a flow-based technique.
Extended abstract. A full version of this paper is [8]. The first author was supported by NSF grant CCF-0915948. The second author was supported by the Barry M. Goldwater Scholarship and by the NSF Graduate Research Fellowship under Grant No. 1122374. The third and fifth authors were supported by DFG grant TH472/4.
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Notes
- 1.
Informally, the value of a game indicates which player will win the game and by how much. A more precise definition, along with much additional background information and full proofs, is given in the full paper [8].
- 2.
These are sometimes called red-blue games in the literature.
- 3.
To clear possible confusion, Col was mistakenly referenced as being proven \(\mathsf{PSPACE}\)-complete in [4].
- 4.
Other variants are possible; see Fukuyama [9] for example, where sticks are placed on edges.
- 5.
Bipartite Geography is one-sided as described above, hence vacuously numeric.
- 6.
If the number of sticks is polynomially bounded, then undirected black-white NimG trivially reduces to undirected Geography and so is clearly in \(\mathsf{P}\).
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We would like to thank an anonymous referee for pointing out that one-sided games, including directed bipartite Geography, are vacuously numeric.
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Fenner, S.A., Grier, D., Messner, J., Schaeffer, L., Thierauf, T. (2015). Game Values and Computational Complexity: An Analysis via Black-White Combinatorial Games. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_58
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