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

2025-07-14

AI Summary

This work studies randomly planted $k$-ary Boolean constraint satisfaction problems (CSPs), aiming to overcome the solvability barrier below the $n^{k/2}$ constraint-density threshold. We propose a family of two-stage algorithms based on Sum-of-Squares (SoS) semidefinite programming relaxations: first, solving a degree-$O(ell)$ SoS relaxation to obtain a vector solution close to the planted assignment; second, applying a novel rounding procedure to exactly recover the planted solution. Crucially, our approach avoids reliance on uniqueness proofs. It succeeds with high probability when the number of constraints satisfies $m geq ilde{O}(n cdot (n/ell)^{k/2-1})$, in runtime $n^{O(ell)}$. Our results unify and generalize prior work by [FPS15] and [RRS17], and—critically—establish, for the first time in the sub-threshold regime, a constraint–time trade-off matching that of refuting random CSPs. The framework applies to arbitrary $k$-ary Boolean CSPs under planted distributions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $x^*$, and then (2) sampling constraints from a particular "planting distribution" designed so that $x^*$ will satisfy every constraint. Given an $n$ variable instance of a $k$-ary Boolean CSP with $m$ constraints, our algorithm runs in time $n^{O(ell)}$ 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 $m geq ilde{O}(n^{frac{k}{2}})$ constraints by exploiting conditions that allow a basic SDP to recover the planted assignment $x^*$ exactly. Instead, we forego certificates of uniqueness and recover $x^*$ 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 $x^*$, and then we use a second rounding procedure to recover $x^*$ from $hat{x}$.

Problem

• Solving random planted k-ary Boolean CSPs efficiently
• Generalizing poly-time algorithms for larger runtimes
• Recovering planted assignments without uniqueness certificates

Innovation

• Uses degree-O(ℓ) Sum-of-Squares SDP
• Two-step recovery of planted assignment
• Generalizes poly(n)-time algorithm to larger runtimes