Tolerant Testing for Unique Games

2026-05-17

AI Summary

This work addresses the problem of efficiently testing satisfiability of Unique Games instances under fault tolerance, without relying on structural assumptions such as expansion or clusterability. The authors present the first sublinear-query tester for general Unique Games in the adjacency list model, distinguishing between instances where the optimal fraction of violated constraints is below $\varepsilon$ and those where it exceeds $\rho$, with query complexity $\tilde{O}(\sqrt{m}\,\rho^{-13/2} + n\rho^{-2}/\sqrt{m})$. For the special case of bipartite Unique Games with $Q=2$, they exploit the signed graph structure to achieve an improved query complexity of $\tilde{O}(\sqrt{m}/\lambda^2 + n/(\sqrt{m}\,\lambda))$, provided that $\varepsilon \log n \lesssim \lambda^2$, where $\lambda = \rho/(1+\log(1/\rho))$, thereby significantly enhancing both tolerance and efficiency.

Abstract

We give tolerant testers with sublinear query complexity in the adjacency-list model for Unique Games. Prior tolerant testers required structural assumptions such as expansion or clusterability. For Unique Games, the tester distinguishes instances whose optimum fraction of violated constraints is at most $\varepsilon$ from those whose optimum is at least $ρ$, for $0<\varepsilon<ρ<1$, assuming $\varepsilon\log n\lesssimρ^4$. On instances with $n$ vertices and $m$ constraints, it uses $\widetilde O(\sqrt m\,ρ^{-13/2}+nρ^{-2}/\sqrt m)$ queries. We also give a specialized tester for bipartiteness, the $Q=2$ transposition case of Unique Games. Exploiting its signed structure, the tester achieves substantially better tolerance and query-complexity guarantees than the generic Unique Games tester. Writing $λ=ρ/(1+\log(1/ρ))$, the bipartiteness tester distinguishes graphs that can be made bipartite by deleting at most an $\varepsilon$ fraction of edges from graphs in which every bipartition has at least a $ρ$ fraction of edges with both endpoints on the same side, assuming $\varepsilon\log n\lesssimλ^2$, using $\widetilde O(\sqrt m/λ^2+n/(\sqrt m\,λ))$ queries.

Problem

• Tolerant Testing
• Unique Games
• Sublinear Query Complexity
• Constraint Satisfaction
• Bipartiteness

Innovation

• tolerant testing
• Unique Games
• sublinear query complexity
• bipartiteness testing
• adjacency-list model