On the Emergence of Ergodic Dynamics in Unique Games

TL;DR

This paper introduces a dynamical systems approach to study the Unique Games Conjecture, showing that unsatisfiable instances exhibit ergodic behavior and linking instance hardness to invariant measure distribution, supporting the conjecture's validity.

Contribution

It develops a novel dynamical systems framework for analyzing unique games and connects ergodic properties with computational hardness, providing new insights into the UGC.

Findings

Unsatisfiable instances lead to ergodic dynamics.

Instance hardness correlates with the distribution of invariant measures.

Numerical results support the UGC hypothesis.

Abstract

The Unique Games Conjecture (UGC) constitutes a highly dynamic subarea within computational complexity theory, intricately linked to the outstanding P versus NP problem. Despite multiple insightful results in the past few years, a proof for the conjecture remains elusive. In this work, we construct a novel dynamical systems-based approach for studying unique games and, more generally, the field of computational complexity. We propose a family of dynamical systems whose equilibria correspond to solutions of unique games and prove that unsatisfiable instances lead to ergodic dynamics. Moreover, as the instance hardness increases, the weight of the invariant measure in the vicinity of the optimal assignments scales polynomially, sub-exponentially, or exponentially depending on the value gap. We numerically reproduce a previously hypothesized hardness plot associated with the UGC. Our results indicate that the UGC is likely true, subject to our proposed conjectures that link dynamical systems theory with computational complexity.