| Popis: |
In this thesis we add to the body of work on parity games. We start by presenting parity games and explaining the concepts behind them, giving a survey of known algorithms, and show their relationship to other problems. In the second part of the thesis we want to answer the following question: Are there classes of graphs on which we can solve parity games in polynomial time? Tree-width has long been considered the most important connectivity measure of (undirected) graphs, and we give a polynomial algorithm for solving parity games on graphs of bounded tree-width. However tree-width is not the most concise measure for directed graphs, on which the parity games are played. We therefore introduce a new measure for directed graphs called DAG-width. We show several properties of this measure, including its relationship to other measures, and give a polynomial-time algorithm for solving parity games on graphs of bounded DAG-width. In the third part we analyze the strategy improvement algorithm of Vöge and Jurdziński, providing some new results and comments on their algorithm. Finally we present a new algorithm for parity games, in part inspired by the strategy improvement algorithm, based on spines. The notion of spine is a new structural way of capturing the (possible) winning sets and counterstrategies. This notion has some interesting properties, which can give a further insight into parity games. |
