  We study the entropy landscape of solutions for the bicoloring problem in
random graphs, a representative difficult constraint satisfaction problem. Our
goal is to classify which type of clusters of solutions are addressed by
different algorithms. In the first part of the study we use the cavity method
to obtain the number of clusters with a given internal entropy and determine
the phase diagram of the problem, e.g. dynamical, rigidity and SAT-UNSAT
transitions. In the second part of the paper we analyze different algorithms
and locate their behavior in the entropy landscape of the problem. For instance
we show that a smoothed version of a decimation strategy based on Belief
Propagation is able to find solutions belonging to sub-dominant clusters even
beyond the so called rigidity transition where the thermodynamically relevant
clusters become frozen. These non-equilibrium solutions belong to the most
probable unfrozen clusters.
