Sublinear-Space Approximation Algorithms for Max r-SAT

Abstract

In the $\textsc {Max}\ r\text {-}\textsc {SAT}{}$ 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 many 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 $n^{{{\,\mathrm{O}\,}}(1)}$ time while using ${{\,\mathrm{O}\,}}(\log {n})$ 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 $n^{{{\,\mathrm{O}\,}}(r)}$ using ${{\,\mathrm{O}\,}}(r \log {n})$ 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 $\textsc {Max}\ r\text {-}\textsc {SAT}{}$ with planar incidence graphs, we devise a factor-$(1 - \epsilon )$ approximation scheme which computes assignments in time $n^{{{\,\mathrm{O}\,}}(r / \epsilon )}$ and uses $\max \{\sqrt{n} \log {n}, (r / \epsilon ) \log ^2{n}\}$ bits of space.