A threshold for unsatisfiability
Journal of Computer and System Sciences
Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On the satisfiability and maximum satisfiability of random 3-CNF formulas
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Bounding the unsatisfiability threshold of random 3-SAT
Random Structures & Algorithms
Gadgets, Approximation, and Linear Programming
SIAM Journal on Computing
The scaling window of the 2-SAT transition
Random Structures & Algorithms
Random Structures & Algorithms
Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
An Upper Bound for the Maximum Cut Mean Value
WG '97 Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science
A 7/8-Approximation Algorithm for MAX 3SAT?
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Optimal myopic algorithms for random 3-SAT
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Linear phase transition in random linear constraint satisfaction problems
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
MaxSolver: an efficient exact algorithm for (weighted) maximum satisfiability
Artificial Intelligence
Proceedings of the 9th annual conference companion on Genetic and evolutionary computation
MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability
Artificial Intelligence
Max k-cut and approximating the chromatic number of random graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
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With random inputs, certain decision problems undergo a "phase transition". We prove similar behavior in an optimization context.Specifically, random 2-SAT formulas with clause/variable density less than 1 are almost always satisfiable, those with density greater than 1 are almost always unsatisfiable, and the "scaling window" is in the density range 1 ± Θ (n-1/3). We prove a similar phase structure for MAX 2-SAT: for density c cn] -- Θ(1/n); within the scaling window it is [cn] -- Θ(1); and for c 1, it is ¾[cn] + Θ(n). (Our results include further detail.)For random graphs, a maximization version of the giant-component question behaves quite differently from 2-SAT, but MAX CUT behaves similarly.For optimization problems, there is also a natural analog of the "satisfiability threshold conjecture". Although here too it remains just a conjecture, it is possible that optimization problems may prove easier to analyze than their decision analogs, and may help to elucidate them.