On the Usefulness of Promises

TL;DR

This paper investigates the concept of promise-usefulness in Boolean predicates, providing conditions to classify predicates as useful or useless for tractable promise problems, with most predicates of large arity being useless.

Contribution

It offers sufficient conditions for promise-usefulness and promise-uselessness, classifies predicates of small arity, and analyzes the prevalence of useless predicates at large arities.

Findings

Most predicates of large arity are promise-useless.

Provided criteria to determine promise-usefulness and uselessness.

Classified all predicates of arity ≤4 and nearly all of arity 5.

Abstract

A Boolean predicate $A$ is defined to be promise-useful if $\operatorname{PCSP}(A,B)$ is tractable for some non-trivial $B$ and otherwise it is promise-useless. We initiate investigations of this notion and derive sufficient conditions for both promise-usefulness and promise-uselessness (assuming $\text{P} \ne \text{NP}$). While we do not obtain a complete characterization, our conditions are sufficient to classify all predicates of arity at most $4$ and almost all predicates of arity $5$. We also derive asymptotic results to show that for large arities a vast majority of all predicates are promise-useless. Our results are primarily obtained by a thorough study of the "Promise-SAT" problem, in which we are given a $k$-SAT instance with the promise that there is a satisfying assignment for which the literal values of each clause satisfy some additional constraint. The algorithmic results are based on the basic LP + affine IP algorithm of Brakensiek et al. (SICOMP, 2020) while we use a number of novel criteria to establish NP-hardness.