4,605 research outputs found
Hiding solutions in random satisfiability problems: A statistical mechanics approach
A major problem in evaluating stochastic local search algorithms for
NP-complete problems is the need for a systematic generation of hard test
instances having previously known properties of the optimal solutions. On the
basis of statistical mechanics results, we propose random generators of hard
and satisfiable instances for the 3-satisfiability problem (3SAT). The design
of the hardest problem instances is based on the existence of a first order
ferromagnetic phase transition and the glassy nature of excited states. The
analytical predictions are corroborated by numerical results obtained from
complete as well as stochastic local algorithms.Comment: 5 pages, 4 figures, revised version to app. in PR
On the freezing of variables in random constraint satisfaction problems
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 large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
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