A deterministic subexponential algorithm for solving parity games

Marcin Jurdziński, Mike Paterson, Uri Zwick
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matoušek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic
more » ... tial algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above. Another important motivation to study the problem of solving parity games is its intriguing complexity theoretic status: the problem is known to be in NP ∩ co-NP [12] and even in UP ∩ co-UP [19] but, as mentioned, despite considerable efforts of the community [12, 20, 32, 15, 1, 28] no polynomial time algorithm has been found so far. (The complexity class UP, aka unambiguous NP, is defined to contain all problems that can be recognized by an unambiguous non-deterministic polynomial-time Turing machine. A Turing machine is unambiguous if for every input it is has at most one accepting computation. Clearly, the inclusions P ⊆ UP ⊆ NP hold.) Moreover, parity games are polynomial time reducible to mean payoff games [34] , simple stochastic games [6] , and discounted payoff games [6, 34] . A stochastic generalization of parity games was also studied [9, 3]. The problems of solving all those games are in UP ∩ co-UP as well [19, 3] . Condon has shown that simple stochastic games are complete (with respect to log-space reductions) in the class of log-space randomized alternating Turing machines [6] . The task of solving parity, mean payoff, discounted payoff, and simple stochastic games can be also viewed as a search problem: given a game graph, compute optimal strategies for both players. The value functions used in strategy improvement algorithms [7, 25, 32, 1] witness membership of all those optimal strategies search problems in PLS (i.e., the class of polynomial local search problems) [17] . On the other hand, the problem of computing optimal strategies in simple stochastic games can be reduced in polynomial time to solving a P-matrix linear complementarity problem [14, 31, 22] , and to finding a Brouwer fixpoint [18] , and hence it is also in PPAD [29] . It follows that there are polynomial time reductions from the problems of computing optimal strategies in parity, mean payoff, discounted payoff, and simple stochastic games to the problem of finding Nash equilibria in bimatrix games [8, 4] . Let n = |V | and m = |E| be the numbers of vertices and edges of a parity game graph and let d be the number of different priorities assigned to vertices by the priority function p : V → N. For parity games with a small number of priorities, more specifically if d = O(n 1/2 ), the progress-measure lifting algorithm [20] gave, until recently, the best time complexity of O(dm(2n/d) d/2 ). This has been improved by Schewe [30] to O(m(κn/d) d/3 ), where κ ≤ (2e) 3/2 . If d = Ω(n (1/2)+ε ) then the randomized algorithm of Björklund et al. [1] has a better (expected) running time bound of n O(
doi:10.1145/1109557.1109571 fatcat:ra2bysahajbyzigah3aa6o6sum