Geometric properties of satisfying assignments of random ε-1-in-k SAT

We study the geometric structure of the set of solutions of random ǫ-1-in-k SAT problem [2, 15]. For l ≥ 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 ǫ-1-in-k SAT instance are O(log n)-connected, conditional on being satisfying assignments. Also, there exists ǫ0 ∈ (0, ) such that w.h.p. no two satisfying assignments at distance at least ǫ0 · n form a ”hole ” in the set of assignments. We believe that this is true for all ǫ> 0, and thus satisfying assignments of a random 1-in-k SAT instance form a single cluster