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The threshold for random $$k$$-SAT is $$2^k\log 2-O(k)$$. (English) Zbl 1093.68075
Summary: Let $$F_k(n,m)$$ be a random $$k$$-CNF formula formed by selecting uniformly and independently $$m$$ out of all possible $$k$$-clauses on $$n$$ variables. It is well known that if $$r \geq 2^k \log 2$$, then $$F_k(n,rn)$$ is unsatisfiable with probability that tends to 1 as $$n \to \infty$$. We prove that if $$r \leq 2^k \log 2- t_k$$, where $$t_k = O(k)$$, then $$F_k(n,rn)$$ is satisfiable with probability that tends to 1 as $$n \to \infty$$.
Our technique, in fact, yields an explicit lower bound for the random $$k$$-SAT threshold for every $$k$$. For $$k \geq 4$$ our bounds improve all previously known such bounds.

##### MSC:
 68R99 Discrete mathematics in relation to computer science 05C80 Random graphs (graph-theoretic aspects) 06E30 Boolean functions 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q25 Analysis of algorithms and problem complexity 68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
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