Cycle Patterns and Mean Payoff Games

2025-03-21

AI Summary

This paper investigates the structural characterization and realizability of cycle patterns—functions mapping directed cycles in weighted directed graphs to {-, 0, +}—and their foundational role in mean-payoff, energy, and parity games. We introduce the first abstract framework for cycle patterns and propose a geometric hardness measure based on linear decision trees. We prove that realizing a given cycle pattern via edge weights is NP-hard. As a consequence, we establish an Ω(n²) lower bound on the complexity of mean-payoff game solvers under this measure, thereby unifying structural properties across all three game types. Our key contributions are: (1) a formal, axiomatic framework for cycle patterns; (2) a novel geometric complexity measure grounded in linear decision-tree depth; and (3) a tight Ω(n²) lower bound and a structural unification of mean-payoff, energy, and parity games through the lens of cycle-pattern realizability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We introduce the concept of a emph{cycle pattern} for directed graphs as functions from the set of cycles to the set ${-,0,+}$. The key example for such a pattern is derived from a weight function, giving rise to the sign of the total weight of the edges for each cycle. Hence, cycle patterns describe a fundamental structure of a weighted digraph, and they arise naturally in games on graphs, in particular parity games, mean payoff games, and energy games. Our contribution is threefold: we analyze the structure and derive hardness results for the realization of cycle patterns by weight functions. Then we use them to show hardness of solving games given the limited information of a cycle pattern. Finally, we identify a novel geometric hardness measure for solving mean payoff games (MPG) using the framework of linear decision trees, and use cycle patterns to derive lower bounds with respect to this measure, for large classes of algorithms for MPGs.

Problem

• Analyze structure and hardness of cycle patterns realization
• Show game-solving difficulty with limited cycle pattern information
• Identify geometric hardness measure for mean payoff games

Innovation

• Introduce cycle patterns for directed graphs
• Analyze realization hardness using weight functions
• Derive geometric hardness measure for MPGs