发表机构
MIT; The Institute for Advanced Study; TTIC（麻省理工学院; 普林斯顿高等研究院; 东京工业计算中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该工作统一了随机CSP的反驳与搜索算法，证明它们均输出带证书的近似最优解，并针对半随机CSP及强污染模型设计了可证明最优的算法。
AI 中文摘要
平均情况复杂度中的一个经典问题是研究随机约束满足问题（CSP）。随机CSP传统上在两种不同的设置下研究：反驳，其中实例是均匀随机的，因此以高概率不可满足；搜索，其中实例来自植入模型，因此是可满足的。虽然随机CSP的反驳和搜索变体之间没有正式的关系，但已知算法惊人地相似，具有几乎相同的计算阈值。在这项工作中，我们通过证明在两种情况下它们都实现了更强的保证，建立了已知的反驳和搜索算法之间的正式关系：它们输出一个赋值$x$以及一个证书$\pi$，表明$x$满足的约束比例在最优赋值的某个小$\varepsilon$范围内。我们将这种保证称为可证明的$\varepsilon$-最优性。作为一个应用，我们为半随机CSP模型设计了新算法，其中实例超图（或作用域）是随机的，但文字否定模式是对抗性选择的，并且可能依赖于超图。对于此类CSP，我们给出了一系列算法，输出可证明的$\varepsilon$-最优解。我们还研究了强污染模型中的此类半随机CSP，其中对手在看到初始CSP后允许破坏$O(\delta)$比例的约束。对于此类CSP，我们给出了一种算法来输出可证明的$O(\delta)$-最优解。
英文摘要
A classical problem in average-case complexity is the study of random constraint satisfaction problems (CSPs). Random CSPs are traditionally studied in two different settings: refutation, where the instances are uniformly random and thus unsatisfiable with high probability, and search, where the instances are drawn from a planted model so that they are satisfiable. While there is no formal relationship between the refutation and search variants of random CSPs, known algorithms are strikingly similar with near-identical computational thresholds. In this work, we establish a formal relationship between the known algorithms for refutation and search by showing that in either case they achieve a stronger guarantee: they output an assignment $x$ along with a certificate $π$ that the fraction of constraints satisfied by $x$ is within some small $\varepsilon$ of the optimal assignment. We call this guarantee certifiable $\varepsilon$-optimality. As an application, we design new algorithms for a model of semirandom CSPs where the instance hypergraph (or scopes) is random, but the literal negation patterns are adversarially chosen and may depend on the hypergraph. For such CSPs, we give a family of algorithms that output certifiably $\varepsilon$-optimal solutions. We additionally study such semirandom CSPs in the "strong contamination model", where an adversary is allowed to corrupt an $O(δ)$-fraction of constraints after seeing the initial CSP. For such CSPs, we give an algorithm to output a certifiably $O(δ)$-optimal solution.
Comments21 pages
