1,689 research outputs found
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
One-step replica symmetry breaking of random regular NAE-SAT I
In a broad class of sparse random constraint satisfaction problems(CSP), deep
heuristics from statistical physics predict that there is a condensation phase
transition before the satisfiability threshold, governed by one-step replica
symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of
such random CSPs, it was verified \cite{ssz16} that its free energy is
well-defined and the explicit value follows the 1RSB prediction. However, for
any model of sparse random CSP, it has been unknown whether the solution space
indeed condensates on O(1) clusters according to the 1RSB prediction. In this
paper, we give an affirmative answer to this question for the random regular
k-NAE-SAT model. Namely, we prove that with probability bounded away from zero,
most of the solutions lie inside a bounded number of solution clusters whose
sizes are comparable to the scale of the free energy. Furthermore, we establish
that the overlap between two independently drawn solutions concentrates
precisely at two values. Our proof is based on a detailed moment analysis of a
spin system, which has an infinite spin space that encodes the structure of
solution clusters. We believe that our method is applicable to a broad range of
random CSPs in the 1RSB universality class.Comment: The previous version is divided into two parts and this submission is
Part I of a two-paper serie
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
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
We establish the existence of free energy limits for several combinatorial
models on Erd\"{o}s-R\'{e}nyi graph and
random -regular graph . For a variety of models, including
independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy
both at a positive and zero temperature, appropriately rescaled, converges to a
limit as the size of the underlying graph diverges to infinity. In the zero
temperature case, this is interpreted as the existence of the scaling limit for
the corresponding combinatorial optimization problem. For example, as a special
case we prove that the size of a largest independent set in these graphs,
normalized by the number of nodes converges to a limit w.h.p. This resolves an
open problem which was proposed by Aldous (Some open problems) as one of his
six favorite open problems. It was also mentioned as an open problem in several
other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999
(Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob\'{a}s and Riordan
[Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin.
Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on
Discrete Structures (2004) 1-72 Springer].Comment: Published in at http://dx.doi.org/10.1214/12-AOP816 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Local geometry of NAE-SAT solutions in the condensation regime
The local behavior of typical solutions of random constraint satisfaction
problems (CSP) describes many important phenomena including clustering
thresholds, decay of correlations, and the behavior of message passing
algorithms. When the constraint density is low, studying the planted model is a
powerful technique for determining this local behavior which in many examples
has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos, Muller
(2020) showed that for a wide class of models, this description applies up to
the so-called condensation threshold.
Understanding the local behavior after the condensation threshold is more
complex due to long-range correlations. In this work, we revisit the random
regular NAE-SAT model in the condensation regime and determine the local weak
limit which describes a random solution around a typical variable. This limit
exhibits a complicated non-Markovian structure arising from the space of
solutions being dominated by a small number of large clusters, a result
rigorously verified by Nam, Sly, Sohn (2021). This is the first
characterization of the local weak limit in the condensation regime for any
sparse random CSPs in the so-called one-step replica symmetry breaking (1RSB)
class.
Our result is non-asymptotic, and characterizes the tight fluctuation
around the limit. Our proof is based on coupling the local
neighborhoods of an infinite spin system, which encodes the structure of the
clusters, to a broadcast model on trees whose channel is given by the 1RSB
belief-propagation fixed point. We believe that our proof technique has broad
applicability to random CSPs in the 1RSB class.Comment: 43 pages, 2 figure
- β¦