New Deterministic Algorithms for Solving Parity Games

TL;DR

This paper introduces fixed-parameter algorithms for solving parity games efficiently when one player controls only a small subset of nodes, improving upon previous methods especially for games with fewer controlling nodes.

Contribution

The paper presents new fixed-parameter algorithms for bipartite and general parity games, with improved time complexity based on the parameter k, and introduces kernelization and deterministic algorithms for specific cases.

Findings

Solved bipartite parity games in time $k^{O(\sqrt{k})} imes O(n^3)$.

Solved general parity games in time $(p+k)^{O(\sqrt{k})} imes O(pnm)$.

Improved deterministic algorithms for graphs with small average degree.

Abstract

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{\'n}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree.