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Random MAX SAT, random MAX CUT, and their phase transitions

Coppersmith, Don, Gamarnik, David, Taghi Hajiaghayi, Mohammad and Sorkin, Gregory B. ORCID: 0000-0003-4935-7820 (2004) Random MAX SAT, random MAX CUT, and their phase transitions. Random Structures and Algorithms, 24 (4). pp. 502-545. ISSN 1042-9832

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Identification Number: 10.1002/rsa.20015


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 &les; &Eopf; max F &les; m. We prove that for random formulas with m = ⌊cn⌋ clauses: for constants c < 1, &Eopf; max 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.

Item Type: Article
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Additional Information: © 2004 Wiley Periodicals, Inc.
Divisions: Management
Subjects: Q Science > QA Mathematics
Date Deposited: 13 Apr 2011 14:23
Last Modified: 16 May 2024 00:16

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