3,047 research outputs found
Random cubic planar graphs revisited
The goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n verticesPostprint (author's final draft
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Non-degenerated groundstates in the antiferromagnetic Ising model on triangulations
A triangulation is an embedding of a graph into a closed Riemann surface so
that each face boundary is a 3-cycle of the graph. In this work, groundstate
degeneracy in the antiferromagnetic Ising model on triangulations is studied.
We show that for every fixed closed Riemann surface S, there are
vertex-increasing sequences of triangulations of S with a non-degenerated
groundstate. In particular, we exhibit geometrically frustrated systems with a
non-degenerated groundstate.Comment: 11 pages, 9 figure
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs
Consider the following problem. A seller has infinite copies of products
represented by nodes in a graph. There are consumers, each has a budget and
wants to buy two products. Consumers are represented by weighted edges. Given
the prices of products, each consumer will buy both products she wants, at the
given price, if she can afford to. Our objective is to help the seller price
the products to maximize her profit.
This problem is called {\em graph vertex pricing} ({\sf GVP}) problem and has
resisted several recent attempts despite its current simple solution. This
motivates the study of this problem on special classes of graphs. In this
paper, we study this problem on a large class of graphs such as graphs with
bounded treewidth, bounded genus and -partite graphs.
We show that there exists an {\sf FPTAS} for {\sf GVP} on graphs with bounded
treewidth. This result is also extended to an {\sf FPTAS} for the more general
{\em single-minded pricing} problem. On bounded genus graphs we present a {\sf
PTAS} and show that {\sf GVP} is {\sf NP}-hard even on planar graphs.
We study the Sherali-Adams hierarchy applied to a natural Integer Program
formulation that -approximates the optimal solution of {\sf GVP}.
Sherali-Adams hierarchy has gained much interest recently as a possible
approach to develop new approximation algorithms. We show that, when the input
graph has bounded treewidth or bounded genus, applying a constant number of
rounds of Sherali-Adams hierarchy makes the integrality gap of this natural
{\sf LP} arbitrarily small, thus giving a -approximate solution
to the original {\sf GVP} instance.
On -partite graphs, we present a constant-factor approximation algorithm.
We further improve the approximation factors for paths, cycles and graphs with
degree at most three.Comment: Preprint of the paper to appear in Chicago Journal of Theoretical
Computer Scienc
Enumeration of labeled 4-regular planar graphs
© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4-regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic calculus. As a by-product, we can also compute the algebraic generating function counting labeled 3-connected 4-regular planar maps.Peer ReviewedPostprint (author's final draft
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