On Computational Hardness of Multidimensional Subtraction Games

On the Hardness of Simple Stochastic Games

We examine the problem of finding optimal strategies in simple stochastic games, and the equivalent problem of finding stable configurations of MIN/MAX/AVG circuits. The problem can be seen as a nontrivial extension to linear programming, but no polynomial-time algorithm is known, despite significant efforts to find such an algorithm. Our investigation will be concerned with the relative comput...

On the Computational Hardness of Graph Coloring

Ultimately bipartite subtraction games

We introduce the notion of ultimately bipartite impartial games. These are games that are ultimately periodic in the simplest possible manner. We examine ultimately bipartite subtraction games and demonstrate a curious feature: for ‘large’ games it is clear who has the winning position and how the game should be strategically played, but during play, as the game eventually becomes small, it is ...

Aperiodic Subtraction Games

Periodicity is a fundamental property of many combinatorial games. It is sought vigorously, yet remains elusive in important cases, such as for some octal games, notably Grundy’s game. Periodicity is important, because it provides poly-time winning strategies for many games. In particular, subtraction games, impartial and partizan, have been proved to be periodic. Our main purpose here is to ex...

Lift-Off Fellowship report On the periodicity of subtraction games

My PhD thesis is aligned with the field of combinatorial games. A combinatorial game is a two-player game in which the players move alternately with neither skipping nor cooperation. There is no hidden information and no element of luck, and so each player understands well the state of the game after each move. There is typically a finite number of moves, so the game must end. The player who ma...