Cycle Patterns and Mean Payoff Games

TL;DR

Cycle patterns map directed cycles to the set $\{-,0,+\}$ and provide a unifying framework for analyzing weighted digraphs and games on graphs. The paper develops a foundational theory of realizability, parity-realizability, and complexity: realizability with circuit representations is shown to be $\text{coNP}$-complete, parity-realizability is $\text{coNP}$-complete, and distinguishing cycle patterns is $\text{NP}$-complete. It then applies cycle-pattern insights to mean payoff, energy, and parity games, establishing hardness results for solving games when only cycle-pattern information is available and introducing a geometric hardness view via linear decision trees (LDTs). A representability measure and the notion that cycle patterns are the coarsest structural descriptor yield lower bounds explaining why many pseudopolynomial MPG algorithms exhibit exponential behavior on worst-case inputs. The extended cycle-pattern framework and RM-based analysis reveal fundamental limits of preprocessing and algorithmic transfer from parity games to MPGs, outlining rich directions for future work in geometric and polyhedral aspects of game solving.

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.

Figures (10)

• Figure 1: Digraph with a cycle pattern that is not realizable.
• Figure 2: Graph that has a cycle pattern for which only large non-realizability witnesses exist. Edge weights are shown on all the edges that have nonzero weight.
• Figure 3: Complete graph on 4 vertices showing the example for \ref{['thm:non-parity-realizable']}
• Figure 4: Left: graph $G_i$ with priorities $\hat{w}$ on the edges. Right: if we use weights $w$ on $G_i$, the induced cycle pattern needs to equal $\psi^{\hat{w}}$.
• Figure 5: A subgraph of $\widehat{G}$ in the case $k=6,i=3$.

Theorems & Definitions (63)

• Definition 2.1
• Lemma 2.2
• proof
• Example 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Proposition 2.6
• proof
• Theorem 2.7