Optimal Values of Multidimensional Mean-Payoff Games
Résumé
In this paper, we study the set of thresholds that the protag-
onist can force in a zero-sum two-player multidimensional mean-payoff
game. The set of maximal elements of such a set is classically called the
Pareto curve, a classical tool to analyze trade-offs. Indeed, as weights
are given as vectors in multiple dimensions, there can be incomparable
such thresholds, and even an infinite number of incomparable ones. Our
main results are as follow. First, we study the geometry of this set and
show that it can be effectively represented as a finite union of convex sets.
Second, we study the computational complexity of natural associated de-
cision problems. In particular, we show that the we can decide in Σ 2 -P
if this set intersects a convex set of points defined by linear inequations.
We also show that this problem is both NP-hard and coNP-hard. Third,
we show that the Pareto curve can be approximated in polynomial time
for fixed number of dimensions and unary encoding of weights.
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