On the Emergence of Ergodic Dynamics in Unique Games

TL;DR

The work embeds 2-Lin-$k$ instances of the Unique Games Conjecture into a continuous dynamical-systems framework in which problem solutions correspond to stable equilibria of a system with energy $\mathcal{V}(\mathbf{s},\mathbf{a})=\sum_{m} a_m K_m(\mathbf{s})^2$ and $K_m(\mathbf{s})=2^{-k}\prod_{p}(1-c_{mp}\mathbf{s}_p)$. It shows that unsatisfiable instances yield ergodic dynamics with an invariant measure on $\mathcal{C}=[-1,1]^N$, and that the weight of the invariant measure near optimal assignments decays with a gap-dependent exponent, reproducing a hardness landscape consistent with the UGC as the alphabet size $k$ grows. The authors argue that ergodicity, rather than transient chaos, serves as a more faithful indicator of problem hardness and provide a numerical bridge to Barak’s proposed hardness picture. They conclude with three conjectures linking dynamical-systems theory to the UGC and outline future work on bounds, universality, and potential algorithmic insights derived from the dynamical framework.

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.

Figures (10)

• Figure 1: Example trajectories evolving on $\mathcal{C}=[-1,1]^{N}$ for a simple UNSAT instance of the 2-Lin-$k$ system of equation.
• Figure 2: Given any unsat instance of Eqn. \ref{['eqn:2lin']}, any trajectory starting inside an arbitrary domain $\mathcal{D}\subseteq\mathcal{C}$ will eventually escape it.
• Figure 3: Method for generating instances of the unique games with user prescribed number of equations. Nodes are variables in the 2-Lin-$k$ system, and lines denote the equations that relate one variable to another. Every solid line corresponds to a satisfiable equation, whereas, every dotted line corresponds to an unsatisfiable equation. Note that one can force the "unsat" equation to be satisfied, at the expense of converting another "sat" equation to "unsat".
• Figure 4: Percentage of time spent by trajectories in the vicinity of assignments that satisfy at least $\delta$ fraction of equations versus the alphabet size $k$ for an instance with $\epsilon=0.4$. Note that the decay rate captures the hardness of the problem and the transition in scaling the assertion of the unique games conjecture.
• Figure 5: Transition of the scaling exponent from Fig. \ref{['fig:transition']} as a function of the size of the problem ($k$ values) and $\delta$. The colors depict the value of $f(\delta,k)$. Green and yellow colors correspond to subexponential values and red corresponds to exponential values ($f(\delta,k)\geq 1$). Results for $\epsilon=0.4$.

Theorems & Definitions (15)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Conjecture 1
• Conjecture 2
• Conjecture 3
• Lemma 1
• proof