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We introduce and study the random locked constraint satisfaction problems. When increasing the density of constraints, they display a broad clustered phase in which the space of solutions is divided into many isolated points. While the phase diagram can be found easily, these problems, in their clustered phase, are extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. We thus propose new benchmarks of really hard optimization problems and provide insight into the origin of their typical hardness.
The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a
We study the phase diagram and the algorithmic hardness of the random `locked constraint satisfaction problems, and compare them to the commonly studied non-locked problems like satisfiability of boolean formulas or graph coloring. The special proper
Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function
Message passing algorithms have proved surprisingly successful in solving hard constraint satisfaction problems on sparse random graphs. In such applications, variables are fixed sequentially to satisfy the constraints. Message passing is run after e
We determine the complexity of several constraint satisfaction problems using the heuristic algorithm, WalkSAT. At large sizes N, the complexity increases exponentially with N in all cases. Perhaps surprisingly, out of all the models studied, the har