The complexity of promise SAT on non-Boolean domains

TL;DR

This paper explores the computational complexity of promise SAT problems on non-Boolean domains, establishing a dichotomy similar to classical SAT results and introducing new algebraic techniques for hardness proofs.

Contribution

It extends the complexity classification of promise SAT problems to arbitrary finite domains and introduces a novel algebraic NP-hardness criterion based on polymorphisms.

Findings

Dichotomy for promise SAT on finite domains analogous to Boolean case

New algebraic NP-hardness criterion based on polymorphisms

Benchmark for algebraic techniques in hardness of approximation

Abstract

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and H\r{a}stad roved a result known as "$(2+\varepsilon)$-SAT is NP-hard" [FOCS'14/SICOMP'17]. They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if $\frac{g}{k} < \frac{1}{2}$ and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains. The hardness side is proved using the algebraic approach, via a new general NP-hardness criterion on polymorphisms of the problem, based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient -- the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation problems such as PCSPs.