Algorithms for lattice games
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This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games (Guo et al. Oberwolfach Rep 22: 23–26, 2009; Guo and Miller, Adv Appl Math 46:363–378, 2010). Given a rational strategy for a lattice game, a polynomial time algorithm is presented to decide (i) whether a given position is a winning position, and to find a move to a winning position, if not; and (ii) to decide whether two given positions are congruent, in the sense of misère quotient theory (Plambeck, Integers, 5:36, 2005; Plambeck and Siegel, J Combin Theory Ser A, 115: 593–622, 2008). The methods are based on the theory of short rational generating functions (Barvinok and Woods, J Am Math Soc, 16: 957–979, 2003).
KeywordsCombinatorial game Lattice game Convex polyhedron Generating function Affine semigroup Misère quotient
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- Barvinok A (2006) The complexity of generating functions for integer points in polyhedra and beyond. International Congress of Mathematicians, vol III. European Mathematical Society, Zurich, pp 763–787Google Scholar
- Bouton CL (1901/02) Nim, a game with a complete mathematical theory. Ann Math (2) 3(1–4):35–39Google Scholar
- Dawson T (1934) Fairy chess supplement. The problemist: British Chess Problem Society 2(9), 94, problem no. 1603Google Scholar
- Guo A, Miller E, Weimerskirch M (2009) Potential applications of commutative algebra to combinatorial game theory. In: Kommutative Algebra, abstracts from the April 19–25, 2009 workshop, organized by W. Bruns, H. Flenner, and C. Huneke, Oberwolfach Rep 22:23–26Google Scholar
- Miller E (2010) Affine stratifications from finite misère quotients (preprint, 2010. arXiv:math. CO/1009. 2199v1) Google Scholar
- Miller E, Sturmfels B (2005) Combinatorial commutative algebra. Graduate texts in mathematics, vol 227. Springer–Verlag, New YorkGoogle Scholar
- Plambeck TE (2005) Taming the wild in impartial combinatorial games. Integers 5(1), G5, p 36 (electronic)Google Scholar
- Plambeck TE, Siegel AN (2008) Misère quotients for impartial games. J Combin Theory Ser A 115(4): 593–622 (arXiv:math.CO/0609825v5) Google Scholar
- Weimerskirch M (2009) An algorithm for computing indistinguishability quotients in misère impartial combinatorial games (preprint)Google Scholar