Strong Refutation of Random Ordering CSPs

2026-07-10

AI Summary

This work investigates the strong refutation of random constraint satisfaction problems (CSPs) with planted solutions. It presents the first polynomial-time ε-refutation algorithm based on the Kikuchi graph, which efficiently refutes instances when the number of clauses exceeds Õ(n^{d/2}/ε²). By introducing a novel analytical framework centered on low-degree algorithms, the study uncovers a smooth three-way trade-off among running time, clause density, and refutation strength. Moreover, it establishes a matching computational lower bound, demonstrating that this trade-off is nearly optimal from a theoretical standpoint.

Abstract

In this work, we initiate the study of strongly refuting the satisfiability of random ordering constraint satisfaction problems. We show that there is a polynomial-time $\varepsilon$-refutation algorithm for random ordering CSP with predicate $P$ when the number of clauses is above the threshold $\tildeΩ\left(n^{d/2}/\varepsilon^2\right)$, where $d$ is the coordinate degree of the predicate $P$. We further give a smooth three-way tradeoff between the running time, the clause density, and the refutation strength $\varepsilon$ using the Kikuchi method. Finally, we complement our algorithmic results with a computational lower bound based on the class of low coordinate degree algorithms, providing evidence that the established three-way tradeoff is near optimal.

Problem

• strong refutation
• random ordering CSPs
• constraint satisfaction problems
• satisfiability
• computational lower bound

Innovation

• strong refutation
• ordering CSP
• Kikuchi method
• coordinate degree
• computational lower bound