The Complexity of Multi-Mean-Payoff and Multi-Energy Games

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. These games have applications in the synthesis of resource-bounded processes with multiple resources. We prove the finite-memory determinacy of multi-energy games and show the inter-reducibility of multimean-payoff and multi-energy games for finite-memory strategies. We also improve the computational complexity for solving both classes of games with finite-memory strategies: while the previously best known upper bound was EXPSPACE, and no lower bound was known, we give an optimal coNP-complete bound. For memoryless strategies, we show that the problem of deciding the existence of a winning strategy for the protagonist is NP-complete. Finally we present the first solution of multi-meanpayoff games with infinite-memory strategies. We show that multi-mean-payoff games with mean-payoff-sup objectives can be decided in NP and coNP, whereas multi-mean-payoff games with mean-payoff-inf objectives are coNP-complete.