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Random MAX SAT, random MAX CUT, and their phase transitions. (English) Zbl 1077.68118
Summary: 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 ifmax \(F\) is equal to \(m\).) With the formula \(F\) chosen at random, the expectation of max \(F\) is trivially bounded by \((3/4)m \leqslant \mathbb E \max F \leqslant m\). We prove that for random formulas with \(m = \lfloor cn \rfloor\) clauses: for constants \(c < 1\), \(\mathbb E \max F\) is \(\lfloor cn \rfloor - \Theta (1/n)\); for large \(c\), it approaches \(((3/4)c + \Theta (\sqrt c))n\); and in the window \(c = 1 + \Theta (n^{-1/3})\), it is \(cn - \Theta (1)\). 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 resultsare 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.

MSC:
68W20 Randomized algorithms
68Q25 Analysis of algorithms and problem complexity
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