布尔CSP反驳的算法与证书：“平滑并不比随机更难”
Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"
AI总结：
本文提出一种算法，强反驳所有布尔CSP的平滑实例，其运行时间与约束数量的权衡匹配随机情形，并利用超图偶覆盖解决了Feige 2008猜想，证明低于谱阈值时存在多项式大小反驳见证。
摘要：
我们提出了一种算法，用于强反驳所有布尔CSP的平滑实例。平滑模型是介于最坏情况和平均情况输入模型之间的混合模型，其中输入是CSP的任意实例，仅字面量的否定模式以某个小概率被重新随机化。对于一个包含n个变量的k元CSP的平滑实例，我们的算法运行时间为n^{O(\ell)}，并且在高概率下成功地将可满足约束的最优比例限制在远离1的范围内，前提是约束数量至少为\tilde{O}(n) (n/\ell)^{k/2 - 1}。这与目前最先进的用于反驳完全随机CSP实例的算法[RRS17]在运行时间和约束数量之间的权衡相匹配，至多相差n的多对数因子。我们还发现了我们的算法与超图中偶覆盖之间的一个令人惊讶的新联系，利用这一联系我们正面解决了Feige 2008年的猜想，这是一个关于足够稠密超图中偶覆盖存在的极值组合学猜想，它推广了著名的图围长Moore界。作为推论，我们证明了对于任意平滑CSP实例，当约束数量比“谱阈值”n^{k/2}低一个多项式因子时，存在多项式大小的反驳见证，这扩展了Feige、Kim和Ofek [FKO06]关于随机3-SAT的著名结果。
英文摘要：
We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of the literals re-randomized with some small probability. For an $n$-variable smoothed instance of a $k$-arity CSP, our algorithm runs in $n^{O(\ell)}$ time, and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from $1$, provided that the number of constraints is at least $\tilde{O}(n) (\frac{n}{\ell})^{\frac{k}{2} - 1}$. This matches, up to polylogarithmic factors in $n$, the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [RRS17]. We also make a surprising new connection between our algorithm and even covers in hypergraphs, which we use to positively resolve Feige's 2008 conjecture, an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the "spectral threshold" of $n^{k/2}$, extending the celebrated result for random 3-SAT of Feige, Kim and Ofek [FKO06].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
