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Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"
arXiv - CS - Computational Complexity Pub Date : 2021-09-09 , DOI: arxiv-2109.04415
Venkatesan Guruswami, Pravesh K. Kothari, Peter Manohar

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].

中文翻译:

布尔 CSP 反驳的算法和证书:“平滑并不比随机难”

我们提出了一种算法来强烈反驳所有布尔 CSP 的平滑实例。平滑模型是最坏情况和平均情况输入模型之间的混合体,其中输入是 CSP 的任意实例,只有文字的否定模式以一些小概率重新随机化。对于 $k$-arity CSP 的 $n$-variable 平滑实例,我们的算法在 $n^{O(\ell)}$ 时间内运行,并且很有可能成功地将可满足约束的最佳部分从$1$,前提是约束的数量至少为 $\tilde{O}(n) (\frac{n}{\ell})^{\frac{k}{2} - 1}$。这匹配,高达 $n$ 中的多对数因子,运行时间和最先进算法的约束数量之间的权衡,用于反驳 CSP 的完全随机实例 [RRS17]。我们还在我们的算法和超图中的偶数覆盖之间建立了一个令人惊讶的新联系,我们用它来肯定地解决 Feige 2008 年的猜想,这是一个关于在足够密集的超图中存在偶数覆盖的极值组合猜想,它概括了众所周知的摩尔界图形的周长。作为推论,我们证明多项式大小的反驳见证存在于任意平滑的 CSP 实例中,约束的数量是多项式因子低于 $n^{k/2}$ 的“光谱阈值”,扩展了随机 3-的著名结果- Feige、Kim 和 Ofek 的 SAT [FKO06]。关于在足够密集的超图中存在偶数覆盖的极值组合猜想,它概括了众所周知的图周长的摩尔边界。作为推论,我们证明多项式大小的反驳见证存在于任意平滑的 CSP 实例中,约束的数量是多项式因子低于 $n^{k/2}$ 的“光谱阈值”,扩展了随机 3-的著名结果- Feige、Kim 和 Ofek 的 SAT [FKO06]。关于在足够密集的超图中存在偶数覆盖的极值组合猜想,它概括了众所周知的图周长的摩尔边界。作为推论,我们证明多项式大小的反驳见证存在于任意平滑的 CSP 实例中,约束的数量是多项式因子低于 $n^{k/2}$ 的“光谱阈值”,扩展了随机 3-的著名结果- Feige、Kim 和 Ofek 的 SAT [FKO06]。
更新日期:2021-09-10
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