[Paper Review] Algorithms for lattice games

Alan Guo, Ezra Miller|arXiv (Cornell University)|May 26, 2011

Advanced Combinatorial Mathematics5 references4 citations

TL;DR

This paper presents polynomial-time algorithms for solving impartial lattice games using rational strategies and affine stratifications, leveraging short rational generating functions. It enables efficient determination of winning positions and misère congruence between positions, offering a computational framework for combinatorial games with bounded complexity in fixed dimensions.

ABSTRACT

This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games. 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. The methods are based on the theory of short rational generating functions.

Motivation & Objective

• To develop effective computational methods for impartial finite combinatorial games formulated as lattice games.
• To enable polynomial-time decision of whether a given position is winning or losing using rational strategies.
• To provide an efficient algorithm for testing misère congruence between positions in lattice games.
• To establish that affine stratifications can be computed in polynomial time, reducing the complexity of strategy computation.
• To support the conjecture that squarefree lattice games possess affine stratifications, advancing the study of misère play in games like Dawson’s Chess.

Proposed method

• Utilizes short rational generating functions to encode winning positions as rational strategies, enabling efficient computation.
• Employs Hadamard products of generating functions to determine reachability of winning positions in polynomial time.
• Applies Barvinok and Woods' theory of short rational generating functions to compute generating functions for affine semigroups and their unions.
• Uses the transformation $ f(S_{ extbf{p}}; \mathbf{t}) = \mathbf{t}^{-\textbf{p}} f((\textbf{p}+C)\cap\mathcal{P}; \mathbf{t}) $ to compute shifted sets for misère congruence testing.
• Applies Corollary 5.3 to combine disjoint rational generating functions of affine stratifications into a single rational function.
• Employs embedding techniques to map affine semigroups into $ \mathbb{N}^d $ for algorithmic tractability, followed by inverse transformation.

Experimental results

Research questions

• RQ1 Can rational strategies be used to decide in polynomial time whether a given position in a lattice game is winning or losing?
• RQ2 Is there an efficient algorithm to test whether two positions are misère congruent, based on their local game structure?
• RQ3 Can affine stratifications be computed in polynomial time, given a lattice game with fixed rule set complexity?
• RQ4 Do squarefree lattice games—such as Dawson’s Chess—admit affine stratifications, enabling efficient strategy computation in misère play?
• RQ5 To what extent do misère quotients and rational strategies remain effective in the presence of aperiodic P-position sets?

Key findings

• Given a rational strategy, there exists a polynomial-time algorithm to determine whether a position is winning or losing, and to compute a move to a winning position if needed.
• Misère congruence between two positions $ \mathbf{p} $ and $ \mathbf{q} $ can be tested in polynomial time by comparing the rational generating functions of the sets $ S_{\mathbf{p}} = (\mathbf{p}+C)\cap\mathcal{P} - \mathbf{p} $ and $ S_{\mathbf{q}} $, with equality implying congruence.
• An affine stratification of a lattice game can be transformed into a rational strategy in polynomial time, assuming fixed complexity of the rule set.
• The complement of a set with an affine stratification also admits an affine stratification, a result of independent interest (Theorem 4.2).
• For a fixed dimension and bounded rule set complexity, the time complexity of computing rational strategies is polynomial in the input size.
• The paper establishes that the generating function of an affine semigroup can be computed in $ O(\iota^c) $ time for some constant $ c $, given complexity $ \iota $, enabling efficient algorithmic manipulation.