Energy Parity Games

TL;DR

Energy parity games combine qualitative parity conditions with quantitative energy constraints on weighted graphs, leading to complex strategies and decision problems that are solvable within NP ∩ coNP, with implications for resource-constrained system design.

Contribution

The paper introduces exponential memory requirements, NP ∩ coNP decision procedures, and reductions to energy games for energy parity games, advancing understanding of their computational complexity.

Findings

Deciding the winner is in NP ∩ coNP.

Exponential memory is necessary and sufficient for winning strategies.

The problem is polynomially equivalent to mean-payoff parity games.

Abstract

Energy parity games are infinite two-player turn-based games played on weighted graphs. The objective of the game combines a (qualitative) parity condition with the (quantitative) requirement that the sum of the weights (i.e., the level of energy in the game) must remain positive. Beside their own interest in the design and synthesis of resource-constrained omega-regular specifications, energy parity games provide one of the simplest model of games with combined qualitative and quantitative objective. Our main results are as follows: (a) exponential memory is necessary and sufficient for winning strategies in energy parity games; (b) the problem of deciding the winner in energy parity games can be solved in NP \cap coNP; and (c) we give an algorithm to solve energy parity by reduction to energy games. We also show that the problem of deciding the winner in energy parity games is polynomially equivalent to the problem of deciding the winner in mean-payoff parity games, while optimal strategies may require infinite memory in mean-payoff parity games. As a consequence we obtain a conceptually simple algorithm to solve mean-payoff parity games.