The computational complexity of some games and puzzles with theoretical applications.

摘要

The subject of this thesis is the algorithmic properties of one- and two-player games people enjoy playing, such as Sudoku or Chess. Questions asked about puzzles and games in this context are of the following type: can we design efficient computer programs that play optimally given any opponent (for a two-player game), or solve any instance of the puzzle in question?;We examine four games and puzzles and show algorithmic as well as intractability results. First, we study the wolf-goat-cabbage puzzle, where a man wants to transport a wolf, a goat, and a cabbage across a river by using a boat that can carry only one item at a time, making sure that no incompatible items are left alone together. We study generalizations of this puzzle, showing a close connection with the VERTEX COVER problem that implies NP-hardness as well as inapproximability results.;Second, we study the SET game, a card game where the objective is to form sets of cards that match in a certain sense using cards from a special deck. We study single- and multi-round variations of this game and establish interesting connections with other classical computational problems, such as PERFECT MULTI-DIMENSIONAL MATCHING, SET PACKING, INDEPENDENT EDGE DOMINATING SET, and A RC KAYLES. We prove algorithmic and hardness results in the classical and the parameterized sense.;Third, we study the UNO game, a game of colored numbered cards where players take turns discarding cards that match either in color or in number. We extend results by Demaine et. al. (2010 and 2014) that connected one- and two-player generalizations of the game to EDGE HAMILTONIAN PATH and GENERALIZED GEOGRAPHY , proving that a solitaire version parameterized by the number of colors is fixed parameter tractable and that a k-player generalization for k greater or equal to 3 is PSPACE-hard.;Finally, we study the Scrabble game, a word game where players are trying to form words in a crossword fashion by placing letter tiles on a grid board. We prove that a generalized version of Scrabble is PSPACE -hard, answering a question posed by Demaine and Hearn in 2008.