On the random satisfiable 3CNF process

In this work we suggest a new model for generating random satisfiable 3CNF formulas. To generate such formulas – randomly permute all 8 ( n 3 ) possible clauses over the variables x1, . . . , xn, 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). For m = cn, c 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 all satisfying assignments are essentially clustered in one cluster, and all but e−Ω(m/n)n of the variables take the same value in all satisfying assignments. We also describe a polynomial time algorithm that finds whp a satisfying assignment for such formulas. Our result coincides with what is already known about the structure of the solution space of formulas drawn from other distributions over satisfiable 3CNF formulas (for the same order of clause-variable ratio) such as the uniform distribution and the planted one. Random processes with conditioning on a certain property being respected were widely studied in the context of graph problems for properties such as planarity, triangle-freeness, etc. Thus our work is a natural extension of this approach to the satisfiability setting. ? Supported in part by USA-Israel BSF Grant 2002-133, and by grant 526/05 from the Israel Science Foundation. On the random satisfiable 3CNF process 1