Robust Approximation and the Arity Barrier at Width Two

2026-09-29

AI Summary

This study addresses whether unbounded arity in width-two constraint languages still admits robust approximation guarantees independent of arity. To investigate Majority-closed linear constraints, the authors integrate octahedral Sum-of-Squares relaxations, truncated Gröbner bases, Gaussian threshold rounding, and Unique Games Conjecture (UGC) reductions. The work reveals, for the first time, an arity barrier for robust approximation at width two, overcoming the finite-arity assumption underlying the Barto–Kozik theorem. Furthermore, it presents a randomized polynomial-time algorithm achieving an O(√ε log k) fraction of violated constraints and establishes a matching Ω(√ε log k) lower bound under the UGC. These results demonstrate that bounded-width characterizations fail in the unbounded-arity setting.

Abstract

Zwick's algorithm for Horn satisfiability shows that constraints of unbounded arity can admit a robust approximation guarantee independent of the arity. For finite constraint languages, Barto and Kozik proved that robust approximability is characterized by bounded width. We ask whether arity-independent robustness persists beyond width one and show that it fails already at width two. For Majority-closed Boolean linear constraints of maximum arity $k \ge 2$, we give a randomized polynomial-time algorithm that, without knowing $\varepsilon$, violates an expected $O(\sqrt{\varepsilon \log k})$ fraction of the constraint weight on $(1-\varepsilon)$-satisfiable instances. Under the Unique Games Conjecture (UGC), a matching NP-hardness lower bound of $Ω(\sqrt{\varepsilon \log k})$ holds in an explicit parameter regime. Under this assumption, the bounded-width characterization therefore does not extend uniformly to constraints of unbounded arity. The algorithm rounds a degree-eight Sum-of-Squares (SoS) relaxation with a single Gaussian threshold. Since polynomial size alone does not make SoS solvable in polynomial bit complexity, we construct complete truncated Groebner bases for the soft Majority ideal and show that, at every fixed degree $2d$, the relaxation can be optimized to any rational accuracy in polynomial time with exactly feasible solutions, and degree-$2d$ SoS proofs can be found after an additive perturbation at degree at most $4d+10$. The lower bound combines Raghavendra's gap-to-hardness theorem with an integrality gap on a Gaussian star, analyzed via the Isaksson-Mossel theorem that parallel halfspaces maximize the joint membership probability of exchangeable Gaussians.

Problem

• robust approximation
• arity barrier
• bounded width
• constraint satisfaction
• Unique Games Conjecture

Innovation

• Robust Approximation
• Sum-of-Squares Relaxation
• Truncated Gröbner Bases
• Unique Games Conjecture
• Arity Barrier