On the Usefulness of Promises

TL;DR

This work introduces and develops promise-usefulness for Boolean PCSPs, defining promise-useful vs promise-useless predicates under a polynomial-time lens. It reduces promise-usefulness to the complexity of fiPCSP(A, OR) (Promise-SAT) and systematically derives tractability criteria via the BLPAFFINE framework, focusing on five block-symmetric polymorphism families. The authors provide complete classifications for predicates of arity up to 4 and almost complete classifications for arity 5, along with asymptotic results showing that random predicates become promise-useless as arity grows; they also establish a threshold for the applicability of the BLPAFFINE algorithm. Across the hardness side, they develop four robust small-fixing-assignment criteria (including ADA-based refinements) that yield NP-hardness for broad families of PCSPs. Collectively, the paper advances a structured, algebraic understanding of when promise-based relaxations are algorithmically tractable, offering both comprehensive small-arity classifications and deep asymptotic insights, with concrete predicates highlighted for future study.

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.

Figures (2)

• Figure 1: Overview of the conditions required for Theorems \ref{['match+ADA']}, \ref{['invmatch+ADA']}, \ref{['thm:unate+ADA']} and \ref{['thm:split']} that ensure the existence of small fixing assignments.
• Figure 2: Distribution of predicates $A$ of arity $5$ such that $\mathop{\mathrm{fiPCSP}}\nolimits(A, \mathop{\mathrm{\mathsf{OR}}}\nolimits)$ is tractable or NP-hard, by weight. The number of unknown predicates is very small but exists for weights between 6 and 12 (both inclusive), and can be glimpsed as thin red lines at weights $6$, $7$, and $8$.

Theorems & Definitions (146)

• Theorem 1.1: Informal version of \ref{['random useless']}
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10