arXiv subjects
Publications and source records attributed to Kyle Burke.
We show that three placement games, Col, NoGo, and Fjords, are PSPACE-complete on planar graphs. The hardness of Col and Fjords is shown via a reduction from Bounded 2-Player Constraint Logic and NoGo is shown to be hard directly from Col.
Consider QBF, the Quantified Boolean Formula problem, as a combinatorial game ruleset. The problem is rephrased as determining the winner of the game where two opposing players take turns assigning values to boolean variables. In this paper, three common variations of games are applied to create seven new games: whether each player is restricted to where they may play, which values they may set variables to, or the condition they are shooting for at the end of the game. The complexity for determining which player can win is analyzed for all games. Of the seven, two are trivially in P and the other five are PSPACE-complete. These varying properties are common for combinatorial games; reductions from these five hard games can simplify the process for showing the PSPACE-hardness of other games.
We build off the game, NimG to create a version named Neighboring Nim. By reducing from Geography, we show that this game is PSPACE-hard. The games created by the reduction share strong similarities with Undirected (Vertex) Geography and regular Nim, both of which are in P. We show how to construct PSPACE-complete versions with nim heaps *1 and *2. This application of graphs can be used as a form of game sum with any games, not only Nim.
Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function.
We create a new two-player game on the Sperner Triangle based on Sperner's lemma. Our game has simple rules and several desirable properties. First, the game is always certain to have a winner. Second, like many other interesting games such as Hex and Geography, we prove that deciding whether one can win our game is a PSPACE-complete problem. Third, there is an elegant balance in the game such that neither the first nor the second player always has a decisive advantage. We provide a web-based version of the game, playable at: http://cs-people.bu.edu/paithan/spernerGame/ . In addition we propose other games, also based on fixed-point theorems.