35,936 research outputs found
Random Graph Models with Hidden Color
We demonstrate how to generalize two of the most well-known random graph
models, the classic random graph, and random graphs with a given degree
distribution, by the introduction of hidden variables in the form of extra
degrees of freedom, color, applied to vertices or stubs (half-edges). The color
is assumed unobservable, but is allowed to affect edge probabilities. This
serves as a convenient method to define very general classes of models within a
common unifying formalism, and allowing for a non-trivial edge correlation
structure.Comment: 17 pages, 2 figures; contrib. to the Workshop on Random Geometry in
Krakow, May 200
Scale-Free Random SAT Instances
We focus on the random generation of SAT instances that have properties
similar to real-world instances. It is known that many industrial instances,
even with a great number of variables, can be solved by a clever solver in a
reasonable amount of time. This is not possible, in general, with classical
randomly generated instances. We provide a different generation model of SAT
instances, called \emph{scale-free random SAT instances}. It is based on the
use of a non-uniform probability distribution to select
variable , where is a parameter of the model. This results into
formulas where the number of occurrences of variables follows a power-law
distribution where . This property
has been observed in most real-world SAT instances. For , our model
extends classical random SAT instances.
We prove the existence of a SAT-UNSAT phase transition phenomenon for
scale-free random 2-SAT instances with when the clause/variable
ratio is . We also prove that scale-free
random k-SAT instances are unsatisfiable with high probability when the number
of clauses exceeds . %This implies that the SAT/UNSAT
phase transition phenomena vanishes when , and formulas are
unsatisfiable due to a small core of clauses. The proof of this result suggests
that, when , the unsatisfiability of most formulas may be due to
small cores of clauses. Finally, we show how this model will allow us to
generate random instances similar to industrial instances, of interest for
testing purposes
On Topological Minors in Random Simplicial Complexes
For random graphs, the containment problem considers the probability that a
binomial random graph contains a given graph as a substructure. When
asking for the graph as a topological minor, i.e., for a copy of a subdivision
of the given graph, it is well-known that the (sharp) threshold is at .
We consider a natural analogue of this question for higher-dimensional random
complexes , first studied by Cohen, Costa, Farber and Kappeler for
.
Improving previous results, we show that is the
(coarse) threshold for containing a subdivision of any fixed complete
-complex. For higher dimensions , we get that is an
upper bound for the threshold probability of containing a subdivision of a
fixed -dimensional complex.Comment: 15 page
- …