Streaming approximation resistance of every ordering CSP

Abstract

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 $ε>0$, MAS is not $(1/2+ε)$-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.