Looking at Mean-Payoff and Total-Payoff through Windows

TL;DR

This paper explores the complexity of multi-dimensional mean-payoff and total-payoff games, introducing window-based approximations that enable decidability and complexity analysis, revealing significant differences between single and multi-dimensional cases.

Contribution

It introduces window-based approximations for multi-dimensional payoff games, establishing complexity results and undecidability boundaries, and compares these with classical objectives.

Findings

Single-dimensional window problems are polynomial-time decidable.

Multi-dimensional fixed window problems are EXPTIME-complete.

Bounded window existence in multi-dimensional total-payoff games is non-primitive recursive.

Abstract

We consider two-player games played on weighted directed graphs with mean-payoff and total-payoff objectives, two classical quantitative objectives. While for single-dimensional games the complexity and memory bounds for both objectives coincide, we show that in contrast to multi-dimensional mean-payoff games that are known to be coNP-complete, multi-dimensional total-payoff games are undecidable. We introduce conservative approximations of these objectives, where the payoff is considered over a local finite window sliding along a play, instead of the whole play. For single dimension, we show that (i) if the window size is polynomial, deciding the winner takes polynomial time, and (ii) the existence of a bounded window can be decided in NP $\cap$ coNP, and is at least as hard as solving mean-payoff games. For multiple dimensions, we show that (i) the problem with fixed window size is EXPTIME-complete, and (ii) there is no primitive-recursive algorithm to decide the existence of a bounded window.