Efficient Algorithms for Semirandom Planted CSPs at the Refutation Threshold

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 $x<sup>*$, (2) choosing an arbitrary clause structure, and (3) choosing literal negations for each clause from an arbitrary distribution "shifted by $x<sup>*$" so that $x<sup>*$ 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 $\tilde{O}(n<sup>{k/2})$ 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 $O(n<sup>k)$ 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.