On the random satisfiable process

TL;DR

This paper introduces a new process for generating random satisfiable k-CNF formulas and analyzes the structure of their solution space, showing that solutions are typically clustered and nearly all variables are fixed across solutions.

Contribution

The paper presents a novel random process for generating satisfiable formulas and characterizes the typical structure of their solution space, including clustering and variable fixation.

Findings

Solutions form a single cluster in typical formulas.

Almost all variables are fixed across solutions.

A polynomial-time algorithm finds satisfying assignments with high probability.

Abstract

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas -- randomly permute all 2^k\binom{n}{k} possible clauses over the variables x_1, ..., x_n, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order). Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruci\'nski and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties were studied such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting. Our main contribution is as follows. For m \geq cn, c=c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e^{-\Omega(m/n)} n of the variables take the same value in all satisfying assignments. We also describe a polynomial time algorithm that finds with high probability a satisfying assignment for such formulas.