  Discrete combinatorial optimization has 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 impossible to solve in linear time when all
solutions contain frozen variables.
