Max Cut for Random Graphs with a Planted Partition
Combinatorics, Probability and Computing
Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-SAT
Theoretical Computer Science
Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time
Combinatorics, Probability and Computing
MAX k-CUT and approximating the chromatic number of random graphs
Random Structures & Algorithms
On the maximum satisfiability of random formulas
Journal of the ACM (JACM)
Approximating almost all instances of MAX-CUT within a ratio above the håstad threshold
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
On an online random k-SAT model
Random Structures & Algorithms
Local Limit Theorems for the Giant Component of Random Hypergraphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
On the satisfiability threshold of formulas with three literals per clause
Theoretical Computer Science
Local search starting from an LP solution: Fast and quite good
Journal of Experimental Algorithmics (JEA)
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Proceedings of the forty-second ACM symposium on Theory of computing
A tighter upper bound for random MAX 2-SAT
Information Processing Letters
ESA'11 Proceedings of the 19th European conference on Algorithms
The MAX-CUT of sparse random graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Limit theorems for random MAX-2-XORSAT
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
An analysis of one-dimensional schelling segregation
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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With random inputs, certain decision problems undergo a “phase transition.” We prove similar behavior in an optimization context. Given a conjunctive normal form (CNF) formula F on n variables and with m k-variable clauses, denote by max F the maximum number of clauses satisfiable by a single assignment of the variables. (Thus the decision problem k-SAT is to determine if max F is equal to m.) With the formula F chosen at random, the expectation of max F is trivially bounded by (3/4)m ⩽ 𝔼 max F ⩽ m. We prove that for random formulas with m = ⌊cn⌋ clauses: for constants c F is ⌊cn⌋ - Θ(1/n); for large c, it approaches ** equation here ** and in the “window” c = 1 + Θ(n-1/3), it is cn - Θ(1). Our full results are more detailed, but this already shows that the optimization problem MAX 2-SAT undergoes a phase transition just as the 2-SAT decision problem does, and at the same critical value c = 1. Most of our results are established without reference to the analogous propositions for decision 2-SAT, and can be used to reproduce them. We consider “online” versions of MAX 2-SAT, and show that for one version the obvious greedy algorithm is optimal; all other natural questions remain open. We can extend only our simplest MAX 2-SAT results to MAX k-SAT, but we conjecture a “MAX k-SAT limiting function conjecture” analogous to the folklore “satisfiability threshold conjecture,” but open even for k = 2. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. We also prove analogous results for random MAX CUT. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004