On the complexity of heterogeneous multidimensional quantitative games

In this paper, we study two-player zero-sum turn-based games played on a finite multidimensional weighted graph. In recent papers all dimensions use the same measure, whereas here we allow to combine different measures. Such heterogeneous multidimensional quantitative games provide a general and natural model for the study of reactive system synthesis. We focus on classical measures like the Inf, Sup, LimInf, and LimSup of the weights seen along the play, as well as on the window mean-payoff (WMP) measure recently introduced in [6]. This new measure is a natural strengthening of the mean-payoff measure. We allow objectives defined as Boolean combinations of heterogeneous constraints. While multidimensional games with Boolean combinations of mean-payoff constraints are undecidable [21], we show that the problem becomes EXPTIME-complete for heterogeneous measures taken among {WMP, Inf, Sup, LimInf, LimSup} and that exponential memory strategies are sufficient for both players to win. We provide a detailed study of the complexity and the memory requirements when the Boolean combination of the measures is replaced by an intersection. EXPTIME-completeness and exponential memory strategies still hold for the intersection of measures in {WMP, Inf, Sup, LimInf, LimSup}, and we get PSPACE-completeness when WMP measure is no longer considered. To avoid EXPTIMEor PSPACE-hardness, we impose at most one occurrence of WMP measure and one occurrence of Sup measure, and we propose several refinements (on the number of occurrences of the other measures) for which we get P-completeness and lower memory requirements. For the considered classes of games, we also study parameterized complexity.