Zielonka's Recursive Algorithm: dull, weak and solitaire games and tighter bounds

TL;DR

This paper analyzes Zielonka's Recursive algorithm's performance on special classes of parity games, revealing exponential lower bounds for dull and solitaire games, and proposing optimizations for polynomial solutions.

Contribution

It provides new complexity bounds for Zielonka's algorithm on dull, weak, and solitaire games, including exponential lower bounds and optimized polynomial-time solutions.

Findings

Zielonka's algorithm runs in O(d(n + m)) on weak games.

Exponential time is required for dull and solitaire games, with lower bounds of 2^(n/3).

Optimizations enable polynomial-time solutions for all three classes.

Abstract

Dull, weak and nested solitaire games are important classes of parity games, capturing, among others, alternation-free mu-calculus and ECTL* model checking problems. These classes can be solved in polynomial time using dedicated algorithms. We investigate the complexity of Zielonka's Recursive algorithm for solving these special games, showing that the algorithm runs in O(d (n + m)) on weak games, and, somewhat surprisingly, that it requires exponential time to solve dull games and (nested) solitaire games. For the latter classes, we provide a family of games G, allowing us to establish a lower bound of 2^(n/3). We show that an optimisation of Zielonka's algorithm permits solving games from all three classes in polynomial time. Moreover, we show that there is a family of (non-special) games M that permits us to establish a lower bound of 2^(n/3), improving on the previous lower bound for the algorithm.