The Threshold for Random k-SAT is 2^k ln2 - O(k)

Details The Threshold for Random k-SAT is 2^k ln2 - O(k) The Threshold for Random k-SAT is 2^k ln2 - O(k)

2003-05-14

arxiv.org/abs/cs/0305009v2

Computer Science

Computational Complexity

Added figures and explained the intuition behind our approach. Made a correction following comments of Chris Calabro

Scientific paper

Let F be a random k-SAT formula on n variables, formed by selecting uniformly and independently m = rn out of all possible k-clauses. It is well-known that if r>2^k ln 2, then the formula F is unsatisfiable with probability that tends to 1 as n tends to infinity. We prove that there exists a sequence t_k = O(k) such that if r < 2^k ln 2 - t_k, then the formula F is satisfiable with probability that tends to 1 as n tends to infinity. Our technique yields an explicit lower bound for the random k-SAT threshold for every k. For k>3 this improves upon all previously known lower bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94.

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