On the critical exponents of random k-SAT

David B. Wilson

There has been much recent interest in the satisfiability of random Boolean formulas. A random k-SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n→ 1, for 3-SAT the critical ratio is thought to be m/n\approx 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called ν=νk (the smaller the value of $\nu$ the sharper the transition). Experiments have suggested that ν3=1.5± 0.1, ν4=1.25±0.05, ν5=1.1±0.05, ν6=1.05±0.05, and heuristics have suggested that νk→ 1 as k→∞. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well-defined). This result holds for each of the three standard ensembles of random k-SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q-colorability and the appearance of a q-core in a random graph.

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