Optimal Streaming Approximations for all Boolean Max-2CSPs and Max-kSAT

TL;DR

This paper establishes tight bounds on streaming approximation ratios for all Boolean Max-2CSP problems, revealing new optimal ratios and separations in space-efficient approximations.

Contribution

It provides the first tight bounds for all Max-2CSPs in streaming, including specific results for Max-DCUT, Max-2SAT, and Max-kSAT, advancing understanding of their approximability.

Findings

Optimal ratio for Max-DCUT is 4/9.

Max-2SAT tight approximation is √2/2.

Max-kSAT tight approximation is √2/2 for all k≥2.

Abstract

We prove tight upper and lower bounds on approximation ratios of all Boolean Max-2CSP problems in the streaming model. Specifically, for every type of Max-2CSP problem, we give an explicit constant $\alpha$, s.t. for any $\epsilon>0$ (i) there is an $(\alpha-\epsilon)$-streaming approximation using space $O(\log{n})$; and (ii) any $(\alpha+\epsilon)$-streaming approximation requires space $\Omega(\sqrt{n})$. This generalizes the celebrated work of [Kapralov, Khanna, Sudan SODA 2015; Kapralov, Krachun STOC 2019], who showed that the optimal approximation ratio for Max-CUT was $1/2$. Prior to this work, the problem of determining this ratio was open for all other Max-2CSPs. Our results are quite surprising for some specific Max-2CSPs. For the Max-DCUT problem, there was a gap between an upper bound of $1/2$ and a lower bound of $2/5$ [Guruswami, Velingker, Velusamy APPROX 2017]. We show that neither of these bounds is tight, and the optimal ratio for Max-DCUT is $4/9$. We also establish that the tight approximation for Max-2SAT is $\sqrt{2}/2$, and for Exact Max-2SAT it is $3/4$. As a byproduct, our result gives a separation between space-efficient approximations for Max-2SAT and Exact Max-2SAT. This is in sharp contrast to the setting of polynomial-time algorithms with polynomial space, where the two problems are known to be equally hard to approximate. Finally, we prove that the tight streaming approximation for \mksat{} is $\sqrt{2}/2$ for every $k\geq2$.