Generalized Mean-payoff and Energy Games

TL;DR

This paper introduces generalized mean-payoff and energy games with multi-dimensional weights, proving their finite-memory determinacy, establishing inter-reducibility, and improving computational complexity bounds to coNP-complete.

Contribution

It proves finite-memory determinacy for generalized energy games, shows inter-reducibility between mean-payoff and energy games, and refines complexity bounds to coNP-complete.

Findings

Finite-memory strategies suffice for determinacy.

Generalized mean-payoff and energy games are inter-reducible.

Complexity of solving these games is coNP-complete.

Abstract

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. Generalized mean-payoff and 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 generalized energy games and show the inter-reducibility of generalized mean-payoff and 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.