Analytic and Algorithmic Solution of Random
Giorgio Parisi, Riccardo Zecchina · 2002
We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio of clauses to variables less than a threshold c are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below c, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms. We introduce a class of optimization algorithms that can deal with these metastable states; one such algorithm has been tested successfully on the largest existing benchmark of K-satisfiability. The K-satisfiability problem (Ksat) asks whether one can satisfy simultaneously a set of M constraints between N Boolean variables, where each constraint is a clause built as the logical OR involving K variables (or their negations). Ksat is at the core of com