A threshold for unsatisfiability
Journal of Computer and System Sciences
Maximum matchings in sparse random graphs: Karp-Sipser revisited
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
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Random MAX SAT, random MAX CUT, and their phase transitions
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
The Asymptotic Order of the Random k -SAT Threshold
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
A Treshold for Unsatisfiability
MFCS '92 Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Random Structures & Algorithms
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Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from C also uniformly at random. This procedure is repeated m times independently. We are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition, when n ← ∞ and m = cn for a constant c. Namely, there exists a critical value c* such that, when c c*, the problem is feasible or is asymptotically almost feasible, as n ← ∞, but, when c c*, the "distance" to feasibility is at least a positive constant independent of n. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [Ald92], [Ald01], Aldous and Steele [AS03], Steele [Ste02] and martingale techniques. By exploiting a linear programming duality, our theorem implies some results for maximum weight matchings in sparse random graphs G(n, ⌊cn⌋) on n nodes with cn edges, where edges are equipped with randomly generated weights.