Dynamic Complexity of Parity Games with Bounded Tree-Width
Résumé
Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of two-player parity games over graphs of bounded tree-width, where updates may add or delete edges, or change the owner or color of states. We show that this problem is in DynFO (with LOGSPACE precomputation); this is achieved by a reduction to a Dyck-path problem on an acyclic automaton. 1 Introduction Parity games. Games played on graphs are tightly linked with automata and logics. In those games, two players move a token along the edges of a graph, thus forming an infinite sequence of states of the graph; each player tries to make this sequence satisfy her winning condition. Such games provide a rich and powerful formalism for dealing with numerous problems in theoretical computer science, especially for verification and synthesis of reactive systems. There is a wide range of games on graphs: they can be zero-sum (involving two players with opposite objectives) or non-zero-sum (involving several players that may want to collaborate); players may or may not have full information about the game, and/or perfect observation of the state of the game; strategies may be deterministic or randomized; and various kinds of winning objectives can be considered. We focus here on two-player perfect-information turn-based (zero-sum) parity games. In such games, the (finitely many) states of the graph are labelled with natural numbers; along each infinite play, the maximal number that appears infinitely many times determines the winner, one of the player trying to make this number even, and the other trying to make it odd. Those games have been extensively studied [17, 8, 24, 14, 9]: deciding the winner in a parity game admits NP and coNP algorithms, but whether this can be achieved in polynomial time remains one of the foremost open problems in this area. Graphs with bounded tree-width. The tree-width of graphs has been defined by Robertson and Seymour [20] as a measure of the complexity of graphs—more precisely, of how close a graph is to a tree. Many classes of graphs have been shown to have bounded tree-width [2]. Over such graphs, several (NP-)hard problems have can be solved efficiently [2]. Parity games are such an example where bounded tree-width makes algorithms more efficient: they have recently been settled in LOGCFL when the underlying graph has bounded tree-width [10]. Dynamic problems and dynamic complexity. In this paper, we focus on the dynamic complexity of parity games. Dynamic complexity theory aims at developing algorithms that are capable of efficiently updating the output of a problem after a slight change in its * This work is supported by EU under ERC EQualIS (FP7-308087) and STREP Cassting (FP7-601148).