核心刻画 CSP 的单遍流式复杂度

中文摘要 AI 辅助

我们研究了在固定有限关系 $R$ 上，针对三种自然目标的一遍流式复杂度：$\ extsf{SAT}(R)$，即判定所有约束是否可满足；$\ extsf{Min}$-$\ extsf{CSP}(R)$，即近似最小化不满足约束的数量；以及 $\ extsf{Exact-CSP}(R)$，即精确计算满足约束的最大数量。Sharma 和 Velusamy (ESA 2026) 使用非冗余参数 $\ extsf{NRD}_n(R)$ 刻画了带文字 CSP 的可满足性流式复杂度，该参数大致衡量每个约束都独立必要的最大的实例。对于不带文字的最一般设置，他们获得了布尔关系的相应刻画，并展示了将其技术扩展到更大定义域上的障碍。Kol、Paramonov、Saxena 和 Yu (ITCS 2023) 类似地以关系写成多项式时的度数 deg$(R)$ 刻画了布尔 CSP 带文字时 $\ extsf{Exact-CSP}(R)$ 的流式复杂度，而不带文字时，相应的结果仅对 $\ extsf{Max-Cut}$ 已知。对于我们研究的全部三种目标，我们给出了任意有限定义域上未加权 CSP 实例的紧刻画。我们提供的流式算法是确定性的，而相应的界在多项式对数因子内是紧的，即使对随机化算法也是如此。所有三个结果的核心思想是将 $R$ 归约到其核心，即 $R$ 的一个规范子关系，它允许使用能够实现变量硬固定（或固定）的 gadgets。

英文摘要

We study the one-pass streaming complexity of CSPs over a fixed finite relation $R$ under three natural objectives: $\textsf{SAT}(R)$, deciding whether all constraints can be satisfied; $\textsf{Min}$-$\textsf{CSP}(R)$, approximately minimizing the number of unsatisfied constraints; and $\textsf{Exact-CSP}(R)$, exactly computing the maximum number of satisfied constraints. Sharma and Velusamy (ESA 2026) characterized the streaming complexity of satisfiability for CSPs with literals using the non-redundancy parameter $\textsf{NRD}_n(R)$, which roughly measures the largest instance in which every constraint is independently necessary. For the most general setting without literals, they obtained the corresponding characterization for Boolean relations and showed obstacles to extending their techniques to larger domains. Kol, Paramonov, Saxena, and Yu (ITCS 2023) similarly characterized the streaming complexity of $\textsf{Exact-CSP}(R)$ for Boolean CSPs with literals in terms of the degree deg$(R)$ of the relation when written as a polynomial, while without literals, the corresponding result was known only for $\textsf{Max-Cut}$. For all three of the objectives we study, we give tight characterizations for unweighted CSP instances over arbitrary finite domains. The streaming algorithms we provide are deterministic, while the corresponding bounds are tight up to polylogarithmic factors, even against randomized algorithms. The central idea in all three results is a reduction of $R$ to its core, a canonical subrelation of $R$ which admits gadgets that allow for the hard-pinning (or fixing) of variables.

发表机构

• University of California, Berkeley（加州大学伯克利分校）
• Harvard University（哈佛大学）
• University of Michigan, Ann Arbor（密歇根大学安娜堡分校）
• University of Waterloo（滑铁卢大学）

机构由 AI 辅助整理，请以论文原文为准。