发表机构
Scuola Universitaria Professionale della Svizzera Italiana, IDSIA; Università della Svizzera Italiana, IDSIA（瑞士意大利语区应用科学大学，IDSIA; 瑞士意大利语区大学，IDSIA）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明鲁棒逼近的元数独立性在宽度二时失效，给出 Majority 闭布尔线性约束的随机算法及 UGC 下匹配下界，并构造 Groebner 基实现 SoS 松弛的高效求解。
AI 中文摘要
Zwick 针对 Horn 可满足性的算法表明，无界元数的约束可以具有独立于元数的鲁棒逼近保证。对于有限约束语言，Barto 和 Kozik 证明了鲁棒可逼近性由有界宽度刻画。我们提出疑问：独立于元数的鲁棒性是否在宽度一之外仍然成立，并证明它在宽度二时即告失效。对于最大元数为 $k \ge 2$ 的 Majority 闭布尔线性约束，我们给出一个随机多项式时间算法，该算法在不知道 $\varepsilon$ 的情况下，在 $(1-\varepsilon)$-可满足实例上违反期望的 $O(\sqrt{\varepsilon \log k})$ 比例的约束权重。在唯一博弈猜想（UGC）下，一个匹配的 NP 困难下界 $\Omega(\sqrt{\varepsilon \log k})$ 在显式参数区间内成立。在此假设下，有界宽度刻画因此不能均匀地扩展到无界元数的约束。该算法对度八的平方和（SoS）松弛进行舍入，使用单一高斯阈值。由于仅有多项式大小并不能使 SoS 在多项式位复杂度下可解，我们为软 Majority 理想构造了完整的截断 Groebner 基，并证明在任意固定度 $2d$ 下，松弛可以在多项式时间内以任意有理精度优化，且得到精确可行解，并且度 $2d$ 的 SoS 证明可以在度至多 $4d+10$ 的加性扰动后找到。下界结合了 Raghavendra 的间隙到困难性定理与高斯星上的积分间隙，后者通过 Isaksson-Mossel 定理分析，该定理表明平行半空间最大化可交换高斯变量的联合成员概率。
英文摘要
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.
