Improved Complexity Analysis of Quasi-Polynomial Algorithms Solving Parity Games

Abstract

We improve the complexity of solving parity games (with priorities in vertices) for $d=\omega (\log n)$ by a factor of $\varTheta (d^2)$: the best complexity known to date was $\mathcal {O}(mdn^{1{.}45+\log _2(d/\log _2 n)})$, while we obtain $\mathcal {O}(mn^{1{.}45+\log _2(d/\log _2 n)}/d)$, where n is the number of vertices, m is the number of edges, and d is the number of priorities.

We base our work on existing algorithms using universal trees, and we improve their complexity. We present two independent improvements. First, an improvement by a factor of $\varTheta (d)$ comes from a more careful analysis of the width of universal trees. Second, we perform (or rather recall) a finer analysis of requirements for a universal tree: while for solving games with priorities on edges one needs an n-universal tree, in the case of games with priorities in vertices it is enough to use an n/2-universal tree. This way, we allow solving games of size 2n in the time needed previously to solve games of size n; such a change divides the quasi-polynomial complexity again by a factor of $\varTheta (d)$.

P. Parys—Author supported by the National Science Centre, Poland (grant no. 2021/41/B/ST6/03914).