Experimental results on the crossover point in random 3-SAT
Artificial Intelligence - Special volume on frontiers in problem solving: phase transitions and complexity
Sudden emergence of a giant k-core in a random graph
Journal of Combinatorial Theory Series B
A sharp threshold for k-colorability
Random Structures & Algorithms
Typical random 3-SAT formulae and the satisfiability threshold
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
2+p-SAT: relation of typical-case complexity to the nature of the phase transition
Random Structures & Algorithms - Special issue on statistical physics methods in discrete probability, combinatorics, and theoretical computer science
The scaling window of the 2-SAT transition
Random Structures & Algorithms
Almost all graphs with average degree 4 are 3-colorable
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Asymptotic Order of the Random k -SAT Threshold
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
The unsatisfiability threshold revisited
Discrete Applied Mathematics
The probabilistic analysis of a greedy satisfiability algorithm
Random Structures & Algorithms
(1,2)-QSAT: A Good Candidate for Understanding Phase Transitions Mechanisms
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
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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 ≈ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called v = vk (the smaller the value of v the sharper the transition). Experiments have suggested that v3 = 1.5 ± 0.1. v4 = 1.25 ± 0.05, v5 = 1.1 ± 0.05, v6 = 1.05 ± 0.05, and heuristics have suggested that vk → 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.