Sublinear-Space Approximation Algorithms for Max r-SAT

TL;DR

This paper develops deterministic sublinear-space approximation algorithms for Max r-SAT, including classical, streaming, and planar graph cases, achieving near-optimal solutions with minimal memory usage.

Contribution

It introduces the first deterministic sublinear-space algorithms for Max r-SAT, extending classical and streaming approaches, and provides a new approximation scheme for planar instances.

Findings

Classical algorithm implemented in n^{O(1)} time with (log n) bits of space.

Streaming algorithm with approximation ratio √2/2 using O(r log n) bits.

Planar graph instances admit a (1 - ε)-approximation in sublinear space.

Abstract

In the Max $r$-SAT problem, the input is a CNF formula with $n$ variables where each clause is a disjunction of at most $r$ literals. The objective is to compute an assignment which satisfies as many of the clauses as possible. While there are a large number of polynomial-time approximation algorithms for this problem, we take the viewpoint of space complexity following [Biswas et al., Algorithmica 2021] and design sublinear-space approximation algorithms for the problem. We show that the classical algorithm of [Lieberherr and Specker, JACM 1981] can be implemented to run in $n^{O(1)}$ time while using $(\log{n})$ bits of space. The more advanced algorithms use linear or semi-definite programming, and seem harder to carry out in sublinear space. We show that a more recent algorithm with approximation ratio $\sqrt{2}/2$ [Chou et al., FOCS 2020], designed for the streaming model, can be implemented to run in time $n^{O(r)}$ using $O(r \log{n})$ bits of space. While known streaming algorithms for the problem approximate optimum values and use randomization, our algorithms are deterministic and can output the approximately optimal assignments in sublinear space. For instances of Max $r$-SAT with planar incidence graphs, we devise a factor-$(1 - \epsilon)$ approximation scheme which computes assignments in time $n^{O(r / \epsilon)}$ and uses $\max\{\sqrt{n} \log{n}, (r / \epsilon) \log^2{n}\}$ bits of space.