On sketching approximations for symmetric Boolean CSPs

TL;DR

This paper derives explicit sketching approximation ratios for symmetric Boolean Max-CSPs, resolving several specific functions and improving understanding of space complexity bounds in streaming algorithms.

Contribution

It provides closed-form expressions for approximation ratios of multiple symmetric Boolean functions in sketching algorithms, extending prior theoretical results.

Findings

Explicit ratios for odd and even k-AND functions.

Resolved ratios for specific threshold functions.

Identified gaps in existing streaming lower bounds for 3-AND.

Abstract

A Boolean maximum constraint satisfaction problem, Max-CSP($f$), is specified by a predicate $f:\{-1,1\}^k\to\{0,1\}$. An $n$-variable instance of Max-CSP($f$) consists of a list of constraints, each of which applies $f$ to $k$ distinct literals drawn from the $n$ variables. For $k=2$, Chou, Golovnev, and Velusamy [CGV20, FOCS 2020] obtained explicit ratios characterizing the $\sqrt n$-space streaming approximability of every predicate. For $k \geq 3$, Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general dichotomy theorem for $\sqrt n$-space sketching algorithms: For every $f$, there exists $\alpha(f)\in (0,1]$ such that for every $\epsilon>0$, Max-CSP($f$) is $(\alpha(f)-\epsilon)$-approximable by an $O(\log n)$-space linear sketching algorithm, but $(\alpha(f)+\epsilon)$-approximation sketching algorithms require $\Omega(\sqrt{n})$ space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting $\alpha'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}$, we show that for odd $k \geq 3$, $\alpha(k$AND$) = \alpha'_k$, and for even $k \geq 2$, $\alpha(k$AND$) = 2\alpha'_{k+1}$. We also resolve the ratio for the "at-least-$(k-1)$-$1$'s" function for all even $k$; the "exactly-$\frac{k+1}2$-$1$'s" function for odd $k \in \{3,\ldots,51\}$; and fifteen other functions. We stress here that for general $f$, according to [CGSV21], closed-form expressions for $\alpha(f)$ need not have existed a priori. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [CGV20] while simplifying [CGSV21]. Finally, we investigate the $\sqrt n$-space streaming lower bounds in [CGSV21], and show that they are incomplete for $3$AND.