Detection of Planted Solutions for Flat Satisfiability Problems

TL;DR

This paper investigates the challenge of detecting planted solutions in flat satisfiability problems, analyzing their properties, detection limits, and the complexity of related algorithms, including a variant linked to learning parity with noise.

Contribution

It introduces a comprehensive analysis of flat satisfiability detection, including optimal detection rates, algorithmic performance, and a new model connecting to learning parity with noise.

Findings

Optimal detection rates characterized

Algorithmic testing performance analyzed

Light planting model shown to be as hard as learning parity with noise

Abstract

We study the detection problem of finding planted solutions in random instances of flat satisfiability problems, a generalization of boolean satisfiability formulas. We describe the properties of random instances of flat satisfiability, as well of the optimal rates of detection of the associated hypothesis testing problem. We also study the performance of an algorithmically efficient testing procedure. We introduce a modification of our model, the light planting of solutions, and show that it is as hard as the problem of learning parity with noise. This hints strongly at the difficulty of detecting planted flat satisfiability for a wide class of tests.