On the Usefulness of Promises

2025-11-26

AI Summary

This work investigates the “promise-usefulness” of a Boolean predicate $A$ in the promise constraint satisfaction problem $ ext{PCSP}(A,B)$: whether there exists a nontrivial predicate $B$ such that $ ext{PCSP}(A,B)$ is polynomial-time solvable. We provide the first systematic definition and characterization of this property. Our method introduces a novel algorithmic framework combining linear programming (LP) relaxations with affine integer programming, complemented by new techniques for proving NP-hardness. This yields sufficient conditions for both promise-usefulness and promise-uselessness. Assuming $ ext{P} eq ext{NP}$, we fully classify all Boolean predicates of arity at most four and handle the vast majority of arity-five predicates. Moreover, we prove that, asymptotically, almost all Boolean predicates of increasing arity are promise-useless. This work establishes the first systematic theory and computational toolkit for determining the tractability boundary of PCSPs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $ ext{P} e ext{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.

Problem

• Classifying Boolean predicates as promise-useful or useless
• Studying Promise-SAT problem with additional clause constraints
• Establishing NP-hardness criteria for promise constraint satisfaction

Innovation

• Defining promise-usefulness for Boolean predicates
• Using LP and affine IP algorithms for tractability
• Applying novel NP-hardness criteria for classification