Because of Cook's theorem, satisfiability lies at the heart of computational complexity theory. This chapter presents some selected research directions, focusing on ensembles of random satisfiability instances. When the density of constraints is increased, a phase transition between a SAT and an UNSAT phase take place. Properly tuned ensembles with a density close to the transition point provide a generator of particularly hard instances. The nature of this transition is discussed, and bounds on the critical density are obtained. On the algorithmic side, the discussion focuses on exhaustive search algorithms based on tree-search, and on random walk procedures.
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