New Algorithms and Hardness Results for Robust Satisfiability of (Promise) CSPs

2026-02-10

AI Summary

This work investigates the computational hardness and algorithmic limits of robustly solving Promise Constraint Satisfaction Problems (PCSPs) under approximate satisfiability. For Boolean and non-Boolean PCSPs admitting algebraic polymorphisms such as Majority or Plurality, it extends algebraic methods to the robust setting for the first time, integrating semidefinite programming (SDP) with correlated rounding, polymorphism analysis, and complexity reductions to establish closure properties of robust satisfiability. The main contributions include proving, under the Unique Games Conjecture (UGC), an exponential loss in approximation for distinguishing 1-in-3-SAT from NAE-SAT, and achieving a robust satisfaction rate of \(1 - O(\sqrt{\varepsilon})\) for PCSPs with Majority polymorphisms—an optimal bound under the UGC. These results reveal fundamental differences in approximability stemming from distinct polymorphism structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• 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

Abstract

In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023 / Discrete Analysis 2025), and obtain the following results: For the PCSP 1-in-3-SAT vs NAE-SAT with negations, we prove that it is hard, under the Unique Games conjecture (UGC), to satisfy $1-\Omega(1/\log (1/\epsilon))$ constraints in a $(1-\epsilon)$-satisfiable instance. This shows that the exponential loss incurred by the BGS algorithm for the case of Alternating-Threshold polymorphisms is necessary, in contrast to the polynomial loss achievable for Majority polymorphisms. For any Boolean PCSP that admits Majority polymorphisms, we give an algorithm satisfying $1-O(\sqrt{\epsilon})$ fraction of the weaker constraints when promised the existence of an assignment satisfying $1-\epsilon$ fraction of the stronger constraints. This significantly generalizes the Charikar--Makarychev--Makarychev algorithm for 2-SAT, and matches the optimal trade-off possible under the UGC. The algorithm also extends, with the loss of an extra $\log (1/\epsilon)$ factor, to PCSPs on larger domains with a certain structural condition, which is implied by, e.g., a family of Plurality polymorphisms. We prove that assuming the UGC, robust satisfiability is preserved under the addition of equality constraints. As a consequence, we can extend the rich algebraic techniques for decision/search PCSPs to robust PCSPs. The methods involve the development of a correlated and robust version of the general SDP rounding algorithm for CSPs due to (Brown-Cohen, Raghavendra, ICALP 2016), which might be of independent interest.

Problem

• Promise CSP
• Robust Satisfiability
• Polymorphisms
• Unique Games Conjecture
• Constraint Satisfaction Problem

Innovation

• Promise CSP
• Robust Satisfiability
• Majority Polymorphism
• Unique Games Conjecture
• SDP Rounding