Alternating Traps in Muller and Parity Games

TL;DR

This paper introduces the concept of trap-depth in Muller and parity games, providing algorithms that efficiently compute winning regions for graphs with bounded trap depth, advancing understanding of game complexity.

Contribution

It defines trap-depth as a new measure of complexity in Muller games and develops polynomial-time algorithms for parity games with bounded trap depth.

Findings

Algorithms run in polynomial time for bounded trap depth

Trap-depth effectively measures game complexity

Exponential time complexity in general cases

Abstract

Muller games are played by two players moving a token along a graph; the winner is determined by the set of vertices that occur infinitely often. The central algorithmic problem is to compute the winning regions for the players. Different classes and representations of Muller games lead to problems of varying computational complexity. One such class are parity games; these are of particular significance in computational complexity, as they remain one of the few combinatorial problems known to be in NP and co-NP but not known to be in P. We show that winning regions for a Muller game can be determined from the alternating structure of its traps. To every Muller game we then associate a natural number that we call its trap-depth; this parameter measures how complicated the trap structure is. We present algorithms for parity games that run in polynomial time for graphs of bounded trap depth, and in general run in time exponential in the trap depth.