The backtracking survey propagation algorithm for solving random K-SAT problems

Discrete combinatorial optimization plays a central role in many scientific disciplines, however for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover it is still unclear what are the key features that make a discrete combinatorial optimization problem hard to solve. Here we study random K-satisfiability problems with K 3, 4 which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. We show that the Backtracking Survey Propagation algorithm, in a time practically linear in the problem size, is able to find solutions very close to the threshold, in a region unreachable by any other algorithm. All solutions found have no frozen variables, thus supporting the conjecture that only unfrozen solutions can be found in linear time, and that a problem becomes impossibile to solve in linear time when all solutions contain frozen variables. Optimization problems with discrete variables are widespread among scientific disciplines and often among the hardest to solve.