Polynomial-Time Algorithms for Energy Games with Special Weight Structures

Abstract

AbstractEnergy games belong to a class of turn-based two-player infinite-duration games played on a weighted directed graph. It is one of the rare and intriguing combinatorial problems that lie in NP ∩ co−NP, but are not known to be in P. While the existence of polynomial-time algorithms has been a major open problem for decades, there is no algorithm that solves any non-trivial subclass in polynomial time.In this paper, we give several results based on the weight structures of the graph. First, we identify a notion of penalty and present a polynomial-time algorithm when the penalty is large. Our algorithm is the first polynomial-time algorithm on a large class of weighted graphs. It includes several counter examples that show that many previous algorithms, such as value iteration and random facet algorithms, require at least sub-exponential time. Our main technique is developing the first non-trivial approximation algorithm and showing how to convert it to an exact algorithm. Moreover, we show that in a practical case in verification where weights are clustered around a constant number of values, the energy game problem can be solved in polynomial time. We also show that the problem is still as hard as in general when the clique-width is bounded or the graph is strongly ergodic, suggesting that restricting graph structures need not help.KeywordsPolynomial TimeEdge WeightIteration AlgorithmFull VersionPositional StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.