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Playing games with algorithms: Algorithmic combinatorial game theory
 IN: PROC. 26TH SYMP. ON MATH FOUND. IN COMP. SCI., LECT. NOTES IN COMP. SCI., SPRINGERVERLAG
, 2001
"... Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, ..."
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Cited by 63 (11 self)
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Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, which analyzes ideal play in perfectinformation games. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomialtime algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.
Generalized Amazons is PSPACE–Complete
"... Amazons is a perfect information board game with simple rules and large branching factors. Two players alternately move chess queenlike pieces and block squares on a 10×10 playing field. The player who makes the last move wins. Amazons endgames usually decompose into independent subgames. Therefore ..."
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Cited by 7 (0 self)
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Amazons is a perfect information board game with simple rules and large branching factors. Two players alternately move chess queenlike pieces and block squares on a 10×10 playing field. The player who makes the last move wins. Amazons endgames usually decompose into independent subgames. Therefore, the game is a natural testbed for combinatorial game theory. It was known that determining the winner of simple generalized Amazons endgames is NPequivalent. This paper presents two proofs for the PSPACEcompleteness of the generalized version of the full game. 1
Hamiltonian Cycles in Triangular Grids
, 2006
"... We study the Hamiltonian Cycle problem in graphs induced by subsets of the vertices of the tiling of the plane with equilateral triangles. By analogy with grid graphs we call such graphs triangular grid graphs. Following the analogy, we define the class of solid triangular grid graphs. We prove that ..."
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Cited by 4 (1 self)
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We study the Hamiltonian Cycle problem in graphs induced by subsets of the vertices of the tiling of the plane with equilateral triangles. By analogy with grid graphs we call such graphs triangular grid graphs. Following the analogy, we define the class of solid triangular grid graphs. We prove that the Hamiltonian Cycle problem is NPcomplete for triangular grid graphs. We show that with the exception of the “Star of David”, a solid triangular grid graph without cut vertices is always Hamiltonian.
Amazons is PSPACEcomplete
, 2005
"... Amazons is a board game which combines elements of Chess and Go. It has become popular in recent years, and has served as a useful platform for both gametheoretic study and AI games research. Buro [2] showed that simple Amazons endgames are NPequivalent, leaving the complexity of the general case ..."
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Cited by 3 (2 self)
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Amazons is a board game which combines elements of Chess and Go. It has become popular in recent years, and has served as a useful platform for both gametheoretic study and AI games research. Buro [2] showed that simple Amazons endgames are NPequivalent, leaving the complexity of the general case as an open problem. We settle this problem, by showing that deciding the outcome of an n×n Amazons position is PSPACEhard. We give a reduction from one of the PSPACEcomplete twoplayer formula games described by Schaefer [9]. Since the number of moves in an Amazons game is polynomially bounded (unlike Chess and Go), Amazons is in PSPACE. It is thus on a par with other twoplayer, boundedmove, perfectinformation games such as Hex [3, 8], Othello [5], and Kayles [9]. Our construction also provides an alternate proof that simple Amazons endgames are NPequivalent. Our reduction uses a number of amazons polynomial in the input formula length; a remaining open problem is the complexity of Amazons when only a constant number of amazons is used. 1
Not being (super)thin or solid is hard: A study of grid Hamiltonicity
, 2008
"... We give a systematic study of Hamiltonicity of grids—the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, “square”, grids, we give a compr ..."
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Cited by 3 (0 self)
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We give a systematic study of Hamiltonicity of grids—the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, “square”, grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomialtime algorithm we give efficient algorithms to find a Hamiltonian cycle. We also establish, for any g ≥ 6, a onetoone correspondence between Hamiltonian cycles in planar bipartite maximumdegree3 graphs and Hamiltonian cycles in the class Cg of girthg planar maximumdegree3 graphs. As applications of the correspondence, we show that for graphs in Cg the Hamiltonian cycle problem is NPcomplete and that for any N ≥ 5 there exist graphs in Cg that have exactly N Hamiltonian cycles. We also prove that for the graphs in Cg, a Chinese Postman tour gives a (1 + 8 g)approximation to TSP, improving thereby the Christofides ratio when g> 16. We show further that, on any graph, the tour obtained by Christofides ’ algorithm is not longer than a Chinese Postman tour.
Experiments in computer amazons
 More Games of No Chance
, 2002
"... Abstract. Amazons is a relatively new game with some similarities to the ancient games of chess and Go. The game has become popular recently with combinatorial games researchers as well as in the computer games community. Amazons combines global fullboard with local combinatorial game features. In ..."
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Cited by 2 (0 self)
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Abstract. Amazons is a relatively new game with some similarities to the ancient games of chess and Go. The game has become popular recently with combinatorial games researchers as well as in the computer games community. Amazons combines global fullboard with local combinatorial game features. In the opening and early middle game, the playing pieces roam freely across the whole board, but later in the game they become confined to one of several small independent areas. A line segment graph is an abstract representation of a local Amazons position. Many equivalent board positions can be mapped to the same graph. We use line segment graphs to efficiently store a table of defective territories, which are important for evaluating endgame positions precisely. We describe the state of the art in the young field of computer Amazons, using our own competitive program Arrow as an example. We also discuss some unusual types of endgame and zugzwang positions that were discovered in the course of writing and testing the program. 1.
Amazons, Konane, and Cross Purposes are PSPACEcomplete
 Games of No Chance 3
, 2007
"... Amazons is a board game which combines elements of Chess and Go. It has become popular in recent years, and has served as a useful platform for both gametheoretic study and AI games research. Buro [3] showed that simple Amazons endgames are NPequivalent, leaving the complexity of the general case ..."
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Cited by 1 (1 self)
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Amazons is a board game which combines elements of Chess and Go. It has become popular in recent years, and has served as a useful platform for both gametheoretic study and AI games research. Buro [3] showed that simple Amazons endgames are NPequivalent, leaving the complexity of the general case as an open problem. 1 Konane is an ancient Hawaiian game, with moves similar to peg solitaire. Konane has received some attention in the combinatorial game theory community, with game values determined for many small positions and onedimensional positions. However, its general complexity seems not to have been previously addressed. Cross Purposes is a game invented by Michael Albert, and named by Richard Guy, at the Games at Dalhousie III workshop, in 2004. It played on a Go board. Cross Purposes is a kind of twoplayer version of the popular puzzle Tipover – the game represents stacks of crates tipping over and blocking others from tipping over. We show that generalized versions of all these games are PSPACEcomplete. We give similar reductions to each game from one of the PSPACEcomplete twoplayer formula games described by Schaefer [17]. Our construction also provides an alternate proof that simple Amazons endgames are NPequivalent. 1
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, 2006
"... There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with twoplayer ga ..."
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There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with twoplayer games. In the first half of this thesis, I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is
An Enhanced Solver for The Game of Amazons
"... Abstract—The game of Amazons is a modern board game with simple rules and nice mathematical properties. It has a high computational complexity. In 2001, the starting position on a 5 × 5 board was proven to be a first player win. The enhanced Amazons solver presented here extends previous work in the ..."
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Abstract—The game of Amazons is a modern board game with simple rules and nice mathematical properties. It has a high computational complexity. In 2001, the starting position on a 5 × 5 board was proven to be a first player win. The enhanced Amazons solver presented here extends previous work in the following five ways: by building more powerful endgame databases, including a new type of databases for socalled blocker territories, by improving the rules for computing bounds on complex game positions, by local search to find tighter local bounds, by using ideas from combinatorial game theory to find wins earlier, and by using a dfpn based solver. Using the improved solver, the starting positions for Amazons on the 4×5, 5×4, 4×6, 5×6 and 4×7 boards were shown to be first player wins, while 6 × 4 is a second player win. The largest proof, for the 5 × 6 board, is presented in detail. I.