105 research outputs found
Satisfiability threshold for random regular NAE-SAT
We consider the random regular -NAE-SAT problem with variables each
appearing in exactly clauses. For all exceeding an absolute constant
, we establish explicitly the satisfiability threshold . We
prove that for the problem is satisfiable with high probability while
for the problem is unsatisfiable with high probability. If the
threshold lands exactly on an integer, we show that the problem is
satisfiable with probability bounded away from both zero and one. This is the
first result to locate the exact satisfiability threshold in a random
constraint satisfaction problem exhibiting the condensation phenomenon
identified by Krzakala et al. (2007). Our proof verifies the one-step replica
symmetry breaking formalism for this model. We expect our methods to be
applicable to a broad range of random constraint satisfaction problems and
combinatorial problems on random graphs
The number of solutions for random regular NAE-SAT
Recent work has made substantial progress in understanding the transitions of
random constraint satisfaction problems. In particular, for several of these
models, the exact satisfiability threshold has been rigorously determined,
confirming predictions of statistical physics. Here we revisit one of these
models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is
natural to study, in the satisfiable regime, the number of solutions in a
typical instance. We prove here that these solutions have a well-defined free
energy (limiting exponential growth rate), with explicit value matching the
one-step replica symmetry breaking prediction. The proof develops new
techniques for analyzing a certain "survey propagation model" associated to
this problem. We believe that these methods may be applicable in a wide class
of related problems
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