Looking at Mean-Payoff and Total-Payoff through Windows
Krishnendu Chatterjee, Laurent Doyen, Mickael Randour and, Jean-Fran\c{c}ois Raskin

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 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…
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Taxonomy
TopicsFormal Methods in Verification · Computability, Logic, AI Algorithms · Complexity and Algorithms in Graphs
