Kolmogorov complexity based upper bounds for the unsatisfiability threshold of random k-SAT

Abstract

We consider Boolean formulas with k literals per clause, or k-SAT formulas. According to the fixed-length random k-SAT model, we can construct a random k-SAT formula by selecting, uniforly at random, m = rn clauses out of the possible ones, for a fixed real number r > 0. As it has been experimentally observed that, as the ratio r of the number m of clauses to the number n of variables is increased from below the approximate value 4.2 to above, then the probability that a random k-SAT formula is satisfiable falls from, asymptotically, 1 to, asymptotically, 0, as n tends to infinity. It is conjectured that a constant value rk exists for all k, called the unsatisfiability threshold for random k-SAT formulas, such that arbitrarily close to it the probabiliy that a random 3-SAT formula is satisfiable falls, abruptly, from 1 to 0. In this paper we deploy techniques from Kolmogorov complexity in order to derive an upper bound to the unsagisfiabliity threshold value generalilzing a similar result from random 3-SAT.