In mean-payoff games, the objective of the protagonist is to ensure that the limit average of an infinite sequence of numeric weights is nonnegative. In energy games, the objective is to ensure that the running sum of weights is always nonnegative. Multi-mean-payoff and multi-energy games replace individual weights by tuples, and the limit average (resp., running sum) of each coordinate must be (resp., remain) nonnegative. We prove finite-memory determinacy of multi-energy games and show inter-reducibility of multi-mean-payoff and multi-energy games for finite-memory strategies. We improve the computational complexity for solving both classes with finite-memory strategies: we prove coNP-completeness improving the previous known EXPSPACE bound. For memoryless strategies, we show that deciding the existence of a winning strategy for the protagonist is NP-complete. We present the first solution of multi-mean-payoff games with infinite-memory strategies: we show that mean-payoff-sup objectives can be decided in NP∩coNP, whereas mean-payoff-inf objectives are coNP-complete.
Information and Computation
The research was partly supported by Austrian Science Fund (FWF) Grant No P23499-N23, FWF NFN Grant No S11407-N23 and S11402-N23 (RiSE), ERC Start grant (279307: Graph Games), Microsoft faculty fellows award, the ERC Advanced Grant QUAREM (267989: Quantitative Reactive Modeling), European project Cassting (FP7-601148), ERC Start grant (279499: inVEST).
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Velner Y, Chatterjee K, Doyen L, Henzinger TA, Rabinovich A, Raskin J. The complexity of multi-mean-payoff and multi-energy games. Information and Computation. 2015;241(4):177-196. doi:10.1016/j.ic.2015.03.001
Velner, Y., Chatterjee, K., Doyen, L., Henzinger, T. A., Rabinovich, A., & Raskin, J. (2015). The complexity of multi-mean-payoff and multi-energy games. Information and Computation. Elsevier. https://doi.org/10.1016/j.ic.2015.03.001
Velner, Yaron, Krishnendu Chatterjee, Laurent Doyen, Thomas A Henzinger, Alexander Rabinovich, and Jean Raskin. “The Complexity of Multi-Mean-Payoff and Multi-Energy Games.” Information and Computation. Elsevier, 2015. https://doi.org/10.1016/j.ic.2015.03.001.
Y. Velner, K. Chatterjee, L. Doyen, T. A. Henzinger, A. Rabinovich, and J. Raskin, “The complexity of multi-mean-payoff and multi-energy games,” Information and Computation, vol. 241, no. 4. Elsevier, pp. 177–196, 2015.
Velner Y, Chatterjee K, Doyen L, Henzinger TA, Rabinovich A, Raskin J. 2015. The complexity of multi-mean-payoff and multi-energy games. Information and Computation. 241(4), 177–196.
Velner, Yaron, et al. “The Complexity of Multi-Mean-Payoff and Multi-Energy Games.” Information and Computation, vol. 241, no. 4, Elsevier, 2015, pp. 177–96, doi:10.1016/j.ic.2015.03.001.