Quiet Planting for $k$-SAT, Multiple Solutions of Arbitrary Geometry

2026-06-14

AI Summary

This work addresses the challenge of “quietly” planting multiple solutions with arbitrary Hamming distances into $k$-SAT instances while preserving computational hardness. The authors introduce a novel planting mechanism that establishes controlled correlations between variable selection and negation patterns, leveraging binary linear $[k, t, r]$ codes to construct distributions satisfying a generalized $(r-1)$-wise uniformity. This approach is the first to enable the planting of an arbitrary number of solutions with prescribed geometric relationships, yielding instances that are statistically indistinguishable from uniformly random ones in the statistical query (SQ) model. Building on this construction, the paper establishes a statistical query lower bound for distinguishing between satisfiable instances with multiple planted solutions and unsatisfiable ones, thereby resolving an open problem posed by Hsieh et al.

Abstract

Recent work on "quiet planting" in combinatorial optimization aims to generate instances with a hidden solution that is hard to recover, typically by making the planted distribution statistically indistinguishable from uniform for specific algorithms, such as statistical queries. A prominent example is planted $k$-SAT, where $O(n^{k/2})$ clauses can be planted while maintaining indistinguishability from uniform instances, evidenced by prior hardness results which also align with findings in SAT refutation. Despite extensive research and practical use in benchmarking SAT solvers, the challenge of quietly planting multiple solutions while preserving hardness has remained an open problem. This work initiates the study of quiet planting with an arbitrary number of solutions, proposing the first method to construct quiet planting distributions for $k$-SAT formulas that accommodate more than one solution. We provide statistical query lower bounds for distinguishing these planted instances from uniform ones, and our method allows for planting solutions with arbitrary geometric relationships, including varying Hamming distances. A key innovation facilitating multiple solutions is the ability to incorporate arbitrary correlations between variable selection in clauses and their negation patterns, departing from prior approaches. We also investigate the worst-case complexity of SAT by showing the difficulty in distinguishing satisfiable instances with numerous solutions from unsatisfiable ones, addressing an open problem of Hsieh, Mohanty, and Xu (CCC'22). Technically, we generalize $(r-1)$-wise uniformness in clause distributions, proving hardness if marginal negation distributions are $(r-1)$-wise uniform. We also reveal a connection to binary linear codes, showing a $[k, t, r]$ code can guide planting up to $2^t - 1$ solutions on $k$ variables.

Problem

• quiet planting
• k-SAT
• multiple solutions
• statistical indistinguishability
• Hamming distance

Innovation

• quiet planting
• k-SAT
• statistical query lower bounds
• multiple solutions
• binary linear codes