4 research outputs found
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
A hard-sphere model on generalized Bethe lattices: Statics
We analyze the phase diagram of a model of hard spheres of chemical radius
one, which is defined over a generalized Bethe lattice containing short loops.
We find a liquid, two different crystalline, a glassy and an unusual
crystalline glassy phase. Special attention is also paid to the close-packing
limit in the glassy phase. All analytical results are cross-checked by
numerical Monte-Carlo simulations.Comment: 24 pages, revised versio
Phase transition in a random minima model: mean field theory and exact solution on the Bethe lattice
We consider the number and distribution of minima in random landscapes
defined on non-Euclidean lattices. Using an ensemble where random landscapes
are reweighted by a fugacity factor for each minimum they contain, we
construct first a `two-box' mean field theory. This exhibits an ordering phase
transition at above which one box contains an extensive number of
minima. The onset of order is governed by an unusual order parameter exponent
, motivating us to study the same model on the Bethe lattice. Here we
find from an exact solution that for any connectivity there is an
ordering transition with a conventional mean field order parameter exponent
, but with the region where this behaviour is observable shrinking
in size as in the mean field limit of large . We show that the
behaviour in the transition region can also be understood directly within a
mean field approach, by making the assignment of minima `soft'. Finally we
demonstrate, in the simplest mean field case, how the analysis can be
generalized to include both maxima and minima. In this case an additional first
order phase transition appears, to a landscape in which essentially all sites
are either minima or maxima.Comment: 31 pages, 3 figure