Efficient Algorithms for Semirandom Planted CSPs at the Refutation Threshold

TL;DR

This paper introduces an efficient polynomial-time algorithm for semirandom planted CSPs that nearly matches the clause threshold of fully random planted CSPs, leveraging spectral techniques to handle worst-case clause structures.

Contribution

The paper presents a novel algorithm that solves semirandom planted CSPs at the refutation threshold, significantly improving over worst-case complexity by exploiting randomness in literal patterns.

Findings

Algorithm runs in polynomial time for n-variable CSPs with O(n^{k/2}) constraints.

Achieves near-complete satisfaction of constraints, missing only o(1) fraction.

Matches clause thresholds of fully random planted CSP algorithms up to polylog factors.

Abstract

We present an efficient algorithm to solve semirandom planted instances of any Boolean constraint satisfaction problem (CSP). The semirandom model is a hybrid between worst-case and average-case input models, where the input is generated by (1) choosing an arbitrary planted assignment $x^*$, (2) choosing an arbitrary clause structure, and (3) choosing literal negations for each clause from an arbitrary distribution "shifted by $x^*$" so that $x^*$ satisfies each constraint. For an $n$ variable semirandom planted instance of a $k$-arity CSP, our algorithm runs in polynomial time and outputs an assignment that satisfies all but a $o(1)$-fraction of constraints, provided that the instance has at least $\tilde{O}(n^{k/2})$ constraints. This matches, up to $polylog(n)$ factors, the clause threshold for algorithms that solve fully random planted CSPs [FPV15], as well as algorithms that refute random and semirandom CSPs [AOW15, AGK21]. Our result shows that despite having worst-case clause structure, the randomness in the literal patterns makes semirandom planted CSPs significantly easier than worst-case, where analogous results require $O(n^k)$ constraints [AKK95, FLP16]. Perhaps surprisingly, our algorithm follows a significantly different conceptual framework when compared to the recent resolution of semirandom CSP refutation. This turns out to be inherent and, at a technical level, can be attributed to the need for relative spectral approximation of certain random matrices - reminiscent of the classical spectral sparsification - which ensures that an SDP can certify the uniqueness of the planted assignment. In contrast, in the refutation setting, it suffices to obtain a weaker guarantee of absolute upper bounds on the spectral norm of related matrices.