Streaming approximation resistance of every ordering CSP

TL;DR

This paper proves that approximating the maximum satisfiability of ordering CSPs in streaming models with sublinear space is fundamentally hard, extending known inapproximability results to all such problems.

Contribution

It establishes the first general streaming inapproximability results for all ordering CSPs, showing they are approximation-resistant in o(n) space, including well-known problems like MAS.

Findings

No o(n)-space streaming algorithm can distinguish nearly satisfiable from unsatisfiable instances.

MAS cannot be approximated better than 1/2 + ε in o(n) space for any ε>0.

The results extend prior work by connecting standard CSP hardness to ordering CSPs via a novel reduction.

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 $\epsilon>0$, 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.