1 research outputs found
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. 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, we
show that the problem becomes EXPTIME-complete for DNF/CNF Boolean combinations
of 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
EXPTIME-or PSPACE-hardness, we impose at most one occurrence of WMP measure and
fix the number of Sup measures, and we propose several refinements (on the
number of occurrences of the other measures) for which we get polynomial
algorithms and lower memory requirements. For all the considered classes of
games, we also study parameterized complexity