Certifiable Near-Optimality: A Simple Framework for Unifying Search and Refutation for (Semi)random CSPs

2026-09-29

AI Summary

This study addresses the longstanding challenge of lacking formal connections between search and refutation algorithms for random constraint satisfaction problems (CSPs). By leveraging average-case complexity theory and semi-random model analysis, this work constructs a unified theoretical framework that rigorously relates search and refutation processes for the first time. It demonstrates that existing algorithms can output near-optimal solutions accompanied by certificates, and proposes optimization methods with verifiable ε-optimality guarantees. Furthermore, novel robust algorithms are designed to operate under strong contamination models. Ultimately, this research achieves verifiably near-optimal solving for semi-random and contaminated CSPs, effectively unifying computational threshold theory and providing a new paradigm that combines theoretical rigor with practical utility for related fields.

Abstract

A classical problem in average-case complexity is the study of random constraint satisfaction problems (CSPs). Random CSPs are traditionally studied in two different settings: refutation, where the instances are uniformly random and thus unsatisfiable with high probability, and search, where the instances are drawn from a planted model so that they are satisfiable. While there is no formal relationship between the refutation and search variants of random CSPs, known algorithms are strikingly similar with near-identical computational thresholds. In this work, we establish a formal relationship between the known algorithms for refutation and search by showing that in either case they achieve a stronger guarantee: they output an assignment $x$ along with a certificate $π$ that the fraction of constraints satisfied by $x$ is within some small $\varepsilon$ of the optimal assignment. We call this guarantee certifiable $\varepsilon$-optimality. As an application, we design new algorithms for a model of semirandom CSPs where the instance hypergraph (or scopes) is random, but the literal negation patterns are adversarially chosen and may depend on the hypergraph. For such CSPs, we give a family of algorithms that output certifiably $\varepsilon$-optimal solutions. We additionally study such semirandom CSPs in the "strong contamination model", where an adversary is allowed to corrupt an $O(δ)$-fraction of constraints after seeing the initial CSP. For such CSPs, we give an algorithm to output a certifiably $O(δ)$-optimal solution.

Problem

• Random CSPs
• Search and Refutation
• Semirandom CSPs
• Certifiable Near-Optimality
• Strong Contamination Model

Innovation

• Certifiable Near-Optimality
• Random CSPs
• Semirandom CSPs
• Search and Refutation
• Strong Contamination Model