On the complexity of heterogeneous multidimensional quantitative games

TL;DR

This paper analyzes the complexity of heterogeneous multidimensional quantitative games, introducing new measures like WMP and exploring their combinations, leading to complexity classifications and strategy memory requirements.

Contribution

It extends the study of multidimensional games by allowing heterogeneous measures and characterizes the complexity and memory needs for various Boolean combinations.

Findings

EXPTIME-complete for certain Boolean combinations of measures

Exponential memory strategies are sufficient for players

Polynomial algorithms exist under specific measure restrictions

Abstract

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.