A Short Implicant of CNFs with Relatively Many Satisfying Assignments

To state our results precisely, we introduce some notation. Throughout this paper, let F be a given Boolean function over N variables, and we assume that it is given as a CNF formula with M clauses and that it has P2N satisfying assignments, where P will be referred as the sat. assignment ratio of F . Furthermore, we introduce two parameters1 δ > 0 and ε > 0, and consider the following situation: (i) P ≥ 2−Nδ , and (ii) M ≤ N1+ε For such a CNF formula F , we discuss the size of its implicant in terms of δ and ε. As our main result, we show that if δ + ε < 1, then one can always find some “short” partial assignment on which F evaluates to 1 by fixing αN variables for some α > 0; that is, F has an implicant of size ≤ αN . (In this paper, for any partial assignment, by its “size” we will mean the number of variables fixed by this assignment; we say that a partial assignment is short if it fixes at most αN variables for some constant α < 1.) If a function F has a short partial assignment, then it has many satisfying assignments. Our result shows that a certain converse relation holds provided that F is expressed as a CNF formula with a relatively small number of clauses. We believe that this structural property would be of some help for designing algorithms for CNF formulas. In fact, we derive, from our analysis, a deterministic algorithm that finds a short partial assignment in Õ(2N β )-time2 for some β < 1 for ∗This work was started from the discussion at the workshop on Computational Complexity at the Banff International Research Station for Mathematical Innovation and Discovery (BIRS), 2013. The first author is supported in part by an NSF postdoctoral research fellowship. The second author is supported in part by the ELC project (MEXT KAKENHI Grant No. 24106008). For simplicity, throughout this paper, we assume that these parameters are constants, and whenever necessary that N is sufficiently large w.r.t. these parameters. By Õ(t(N)) we mean O ( t(N)(log t(N)) ) .