An ordering constraint satisfaction problem (OCSP) is defined by a family $\mathcal{F}$ of predicates mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. An instance of Max-OCSP($\mathcal{F}$) on $n$ variables consists of a list of constraints, each consisting of a predicate from $\mathcal{F}$ applied on $k$ distinct variables. The goal is to find an ordering of the $n$ variables that maximizes the number of constraints for which the induced ordering on the $k$ variables satisfies the predicate. OCSPs capture well-studied problems including `maximum acyclic subgraph' (MAS) and "maximum betweenness". In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every $\mathcal{F}$, Max-OCSP($\mathcal{F}$) is approximation-resistant to $o(n)$-space streaming algorithms, i.e., algorithms using $o(n)$ space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $(1/2+\epsilon)$-approximable in $o(n)$ space. The previous best inapproximability result, due to Guruswami and Tao (APPROX'19), only ruled out $3/4$-approximations in $o(\sqrt n)$ space. Our results build on a recent work of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC'22), who provide a tight, linear-space inapproximability theorem for a broad class of "standard" (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. We construct a family of appropriate standard CSPs from any given OCSP, apply their hardness result to this family of CSPs, and then convert back to our OCSP.

翻译：排序约束满足问题（OCSP）由一族将 $\{1,\ldots,k\}$ 上的排列映射到 $\{0,1\}$ 的谓词 $\mathcal{F}$ 定义。一个包含 $n$ 个变量的 Max-OCSP($\mathcal{F}$) 实例由一组约束列表构成，其中每个约束是将 $\mathcal{F}$ 中的一个谓词作用于 $k$ 个不同变量上。目标是找到 $n$ 个变量的一个排序，使得在该排序下，$k$ 个变量上诱导出的顺序满足谓词的约束数量最大化。OCSP 涵盖了许多被深入研究的经典问题，包括“最大无环子图”（MAS）和“最大中介性”。在本工作中，我们考虑在（单遍）流式设置下近似可满足约束的最大数量，其中实例以约束流的形式呈现。我们证明对于任意 $\mathcal{F}$，Max-OCSP($\mathcal{F}$) 对 $o(n)$ 空间流式算法是近似抵抗的，即使用 $o(n)$ 空间的算法无法区分几乎所有约束均可满足的流与不存在任何排序能显著优于随机排序的流。该空间界限在对数多项式因子内是紧的。对于 MAS 问题，我们的结果表明对于任意 $\epsilon>0$，MAS 无法在 $o(n)$ 空间内实现 $(1/2+\epsilon)$ 近似。此前由 Guruswami 和 Tao（APPROX'19）给出的最佳不可近似性结果仅排除了在 $o(\sqrt n)$ 空间内实现 $3/4$ 近似的可能性。我们的结果建立在 Chou、Golovnev、Sudan、Velingker 和 Velusamy（STOC'22）近期工作的基础上，他们针对任意（有限）字母表上的一大类“标准”（即非排序）约束满足问题（CSP）给出了一个紧的线性空间不可近似性定理。我们从任意给定的 OCSP 构造出一族适当的标准 CSP，将他们的硬度结果应用于该 CSP 族，再转换回我们的 OCSP。