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.

翻译：《独特游戏猜想》（UGC）是计算复杂性理论中一个高度活跃的子领域，与著名的P与NP问题紧密相关。尽管过去几年取得了多项深刻成果，但该猜想的证明仍悬而未决。本文构建了一种新颖的基于动力系统的研究方法，用于研究独特游戏及更广泛的计算复杂性领域。我们提出了一族动力系统，其平衡点对应独特游戏的解，并证明了不可满足实例将导致遍历动力学。此外，随着实例难度的增加，最优解附近不变测度的权重根据值间隙以多项式、次指数或指数方式缩放。我们数值重现了此前假设的与UGC相关的难度图。我们的结果表明，在将动力系统理论与计算复杂性相联系的前提下，UGC很可能是正确的。