New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number \(k\) of nodes and the other player controls the \(n-k\) other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time \(k^{O(\sqrt{k})}\cdot O(n^3)\), and general parity games in time \((p+k)^{O(\sqrt{k})} \cdot O(pnm)\), where \(p\) is the number of distinct priorities and \(m\) is the number of edges. For all games with \(k = o(n)\) this improves the previously fastest algorithm by Jurdzi{ń}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree.