On the parameterized complexity of finding short winning strategies in combinatorial games




Scott, Allan Edward Jolicoeur

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A combinatorial game is a game in which all players have perfect information and there is no element of chance; some well-known examples include othello, checkers, and chess. When people play combinatorial games they develop strategies, which can be viewed as a function which takes as input a game position and returns a move to make from that position. A strategy is winning if it guarantees the player victory despite whatever legal moves any opponent may make in response. The classical complexity of deciding whether a winning strategy exists for a given position in some combinatorial game has been well-studied both in general and for many specific combinatorial games. The vast majority of these problems are, depending on the specific properties of the game or class of games being studied, complete for either PSPACE or EXP. In the parameterized complexity setting, Downey and Fellows initiated a study of "short" (or k-move) winning strategy problems. This can be seen as a generalization of "mate-in-k" chess problems, in which the goal is to find a strategy which checkmates your opponent within k moves regardless of how he responds. In their monograph on parameterized complexity, Downey and Fellows suggested that AW[*] was the "natural home" of short winning strategy problems, but there has been little work in this field since then. In this thesis, we study the parameterized complexity of finding short winning strategies in combinatorial games. We consider both the general and several specific cases. In the general case we show that many short games are as hard classically as their original variants, and that finding a short winning strategy is hard for AW[P] when the rules are implemented as succinct circuits. For specific short games, we show that endgame problems for checkers and othello are in FPT, that alternating hitting set, hex, and the non-endgame problem for othello are in AW[*], and that short chess is AW[*]-complete. We also consider pursuit-evasion parameterized by the number of cops. We show that two variants of pursuit-evasion are AW[*]-hard, and that the short versions of these problems are AW[*]-complete.



computational complexity, combinatorial game theory, parameterized complexity, algorithmic combinatorial game theory