Strongly Refuting Random CSP without Literals

2026-04-29

AI Summary

This work investigates the hardness of refuting random unsatisfiable k-CSPs without literals within the Sum-of-Squares (SoS) framework. We introduce t-wise independence as a necessary and sufficient condition for SoS hardness, extending beyond prior limitations to Boolean domains and uniform literals, and generalize the optimal three-way tradeoff—among constraint density, SoS degree, and refutation strength—to arbitrary domains and non-uniform settings. Key technical contributions include a novel Kikuchi matrix tailored for odd-order and asymmetric tensors, integration with global correlation rounding, and a spectral refutation algorithm that achieves strong refutation for general random k-CSPs without incurring runtime penalties, thereby attaining theoretical optimality.

Abstract

Under what condition is a random constraint satisfaction problem hard to refute by the sum-of-squares (SoS) algorithm? A sufficient condition is t-wise uniformity, that is, each constraint has a t-wise uniform distribution of satisfying assignments, as shown by the lower bounds of Kothari, Mori, O'Donnell, and Witmer (STOC 2017). This condition is also necessary for random CSPs given by a predicate and uniformly random literals, due to the constant-degree SoS refutation of Allen, O'Donnell, and Witmer (FOCS 2015). For higher degree, Raghavendra, Rao, and Schramm (STOC 2017) gave a refutation for Boolean random CSPs with uniformly random literals, matching the lower bounds optimally in terms of the three-way tradeoff between constraint density, SoS degree, and strength of refutation. Two long-standing open problems are to find a more general sufficient condition for SoS lower bounds, and to refute similar random CSPs not involving literals. We show that for a general random k-CSP, the necessary and sufficient hardness condition is not t-wise uniformity, but t-wise independence. We generalize the optimal three-way tradeoff to any random k-CSP, without assuming a Boolean domain or uniformly random literals. Our analysis involves new Kikuchi matrices for odd order and for asymmetric tensors. It also uses the global correlation rounding technique of Barak, Raghavendra, and Steurer (FOCS 2011). To avoid the running-time penalty of this technique, we also give a spectral refutation algorithm.

Problem

• random CSP
• sum-of-squares
• t-wise independence
• refutation
• constraint satisfaction problem

Innovation

• t-wise independence
• sum-of-squares hierarchy
• random CSP
• Kikuchi matrices
• spectral refutation