Solving Random Planted CSPs below the $n^{k/2}$ Threshold

Abstract: We present a family of algorithms to solve random planted instances of any $k$-ary Boolean constraint satisfaction problem (CSP). A randomly planted instance of a Boolean CSP is generated by (1) choosing an arbitrary planted assignment $x<sup>*$, and then (2) sampling constraints from a particular "planting distribution" designed so that $x<sup>*$ will satisfy every constraint. Given an $n$ variable instance of a $k$-ary Boolean CSP with $m$ constraints, our algorithm runs in time $n<sup>{O(\ell)}$ for a choice of a parameter $\ell$, and succeeds in outputting a satisfying assignment if $$m \geq O(n) \cdot (n/\ell)^{\frac{k}{2} - 1} \log n$$. This generalizes the $\mathrm{poly}(n)$-time algorithm of [FPV15], the case of $\ell = O(1)$, to larger runtimes, and matches the constraint number vs.\ runtime trade-off established for refuting random CSPs by [RRS17]. Our algorithm is conceptually different from the recent algorithm of [GHKM23], which gave a $\mathrm{poly}(n)$-time algorithm to solve semirandom CSPs with $m \geq \tilde{O}(n<sup>{\frac{k}{2}})$ constraints by exploiting conditions that allow a basic SDP to recover the planted assignment $x<sup>*$ exactly. Instead, we forego certificates of uniqueness and recover $x<sup>*$ in two steps: we first use a degree-$O(\ell)$ Sum-of-Squares SDP to find some $\hat{x}$ that is $o(1)$-close to $x<sup>*$, and then we use a second rounding procedure to recover $x<sup>*$ from $\hat{x}$.