Geometric properties of satisfying assignments of random $\epsilon$-1-in-k SAT

TL;DR

This paper investigates the geometric structure of solutions in random epsilon-1-in-k SAT problems, showing that solutions are typically well-connected and form a single cluster, with no large gaps between them.

Contribution

It proves that satisfying assignments are logarithmically connected and that large-distance solutions do not create holes, suggesting a single solution cluster.

Findings

Satisfying assignments are O(log n)-connected with high probability.

No large holes exist between solutions at a linear distance, with high probability.

Solutions form a single connected cluster, supporting conjectures about the solution space structure.

Abstract

We study the geometric structure of the set of solutions of random $\epsilon$-1-in-k SAT problem. For $l\geq 1$, two satisfying assignments $A$ and $B$ are $l$-connected if there exists a sequence of satisfying assignments connecting them by changing at most $l$ bits at a time. We first prove that w.h.p. two assignments of a random $\epsilon$-1-in-$k$ SAT instance are $O(\log n)$-connected, conditional on being satisfying assignments. Also, there exists $\epsilon_{0}\in (0,\frac{1}{k-2})$ such that w.h.p. no two satisfying assignments at distance at least $\epsilon_{0}\cdot n$ form a "hole" in the set of assignments. We believe that this is true for all $\epsilon >0$, and thus satisfying assignments of a random 1-in-$k$ SAT instance form a single cluster.