The first cycles in an evolving graph
Discrete Mathematics
A threshold for unsatisfiability
Journal of Computer and System Sciences
Satisfiability threshold for random XOR-CNF formulas
Discrete Applied Mathematics - Special issue on the satisfiability problem and Boolean functions
The scaling window of the 2-SAT transition
Random Structures & Algorithms
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Forbidden subgraphs in connected graphs
Theoretical Computer Science
Random $k$-SAT: Two Moments Suffice to Cross a Sharp Threshold
SIAM Journal on Computing
Another proof of Wright's inequalities
Information Processing Letters
Mick gets some (the odds are on his side) (satisfiability)
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Analytic Combinatorics
Phase transition for random quantified XOR-formulas
Journal of Artificial Intelligence Research
Limit theorems for random MAX-2-XORSAT
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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We consider the random 2-XOR satisfiability problem, in which each instance is a formula that is a conjunction of m Boolean equations of the form x⊕y = 0 or x⊕y = 1. Random formulas on n Boolean variables are chosen uniformly at random from among all (n(n-1) m) possible choices. This problem is known to have a coarse transition as n and m tends to infinity in the ratio m/n → c in particular the probability p(n, cn) that a random 2-XOR formula is satisfiable tends to zero when c reaches c = 1/2. We determine the rate n-1/12 at which this probability approaches zero inside the scaling window m = n/2 (1 + µn-1/3). This main result is based on a first exact enumeration result about the number of connected components of some constrained family of random edge-weighted (0/1) graphs, namely those without cycles of odd weight. This study relies on the symbolic method and analytical tools coming from generating function theory which enable us to describe the evolution of n1/12 p(n, n/2(1 + µn-1/3)) as a function of µ. Our results are in accordance with those obtained by statistical physics methods, their tightness points out the benefit one could get in developping generating function methods for the investigation of phase transition associated to Constrained Satisfaction Problems.