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2003
ACM

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

9 years 3 months ago
Random MAX SAT, random MAX CUT, and their phase transitions
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 < 1, E max F is cn − Θ(1/n); for large c, it approaches (3 4 c + Θ( √ c))n; 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
Don Coppersmith, David Gamarnik, Mohammad Taghi Ha
Added 01 Nov 2010
Updated 01 Nov 2010
Type Conference
Year 2003
Where SODA
Authors Don Coppersmith, David Gamarnik, Mohammad Taghi Hajiaghayi, Gregory B. Sorkin
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