Revisiting Maximum Satisfiability and Related Problems in Data Streams

• Main Author: Vu, Hoa T
• Format: Journal Article
• Language: English
• Published: 19-08-2022
• Subjects: Computer Science - Data Structures and Algorithms
• DOI: 10.48550/arxiv.2208.09160

We revisit the MaxSAT problem in the data stream model. In this problem, the
stream consists of $m$ clauses that are disjunctions of literals drawn from $n$
Boolean variables. The objective is to find an assignment to the variables that
maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that
$\Omega(\sqrt{n})$ space is necessary to yield a $\sqrt{2}/2+\epsilon$
approximation of the optimum value; they also presented an algorithm that
yields a $\sqrt{2}/2-\epsilon$ approximation of the optimum value using $O(\log
n/\epsilon^2)$ space.
In this paper, we focus not only on approximating the optimum value, but also
on obtaining the corresponding Boolean assignment using sublinear $o(mn)$
space. We present randomized single-pass algorithms that w.h.p. yield: 1) A
$1-\epsilon$ approximation using $\tilde{O}(n/\epsilon^3)$ space and
exponential post-processing time and 2) A $3/4-\epsilon$ approximation using
$\tilde{O}(n/\epsilon)$ space and polynomial post-processing time. Our ideas
also extend to dynamic streams. On the other hand, we show that the streaming
kSAT problem that asks to decide whether one can satisfy all size-$k$ input
clauses must use $\Omega(n^k)$ space.
We also consider other related problems in this setting.