37,785 research outputs found
Counting unlabelled toroidal graphs with no K33-subdivisions
We provide a description of unlabelled enumeration techniques, with complete
proofs, for graphs that can be canonically obtained by substituting 2-pole
networks for the edges of core graphs. Using structure theorems for toroidal
and projective-planar graphs containing no K33-subdivisions, we apply these
techniques to obtain their unlabelled enumeration.Comment: 25 pages (some corrections), 4 figures (one figure added), 3 table
Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs
We describe the structure of 2-connected non-planar toroidal graphs with no
K_{3,3}-subdivisions, using an appropriate substitution of planar networks into
the edges of certain graphs called toroidal cores. The structural result is
based on a refinement of the algorithmic results for graphs containing a fixed
K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing
K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these
graphs in linear-time and makes possible to enumerate labelled 2-connected
toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex
degree two or three by using an approach similar to [A. Gagarin, G. Labelle,
and P. Leroux, "Counting labelled projective-planar graphs without a
K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table
An adaptive prefix-assignment technique for symmetry reduction
This paper presents a technique for symmetry reduction that adaptively
assigns a prefix of variables in a system of constraints so that the generated
prefix-assignments are pairwise nonisomorphic under the action of the symmetry
group of the system. The technique is based on McKay's canonical extension
framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the
technique are (i) adaptability---the prefix sequence can be user-prescribed and
truncated for compatibility with the group of symmetries; (ii)
parallelizability---prefix-assignments can be processed in parallel
independently of each other; (iii) versatility---the method is applicable
whenever the group of symmetries can be concisely represented as the
automorphism group of a vertex-colored graph; and (iv) implementability---the
method can be implemented relying on a canonical labeling map for
vertex-colored graphs as the only nontrivial subroutine. To demonstrate the
practical applicability of our technique, we have prepared an experimental
open-source implementation of the technique and carry out a set of experiments
that demonstrate ability to reduce symmetry on hard instances. Furthermore, we
demonstrate that the implementation effectively parallelizes to compute
clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie
A Generalization of the {\L}o\'s-Tarski Preservation Theorem over Classes of Finite Structures
We investigate a generalization of the {\L}o\'s-Tarski preservation theorem
via the semantic notion of \emph{preservation under substructures modulo
-sized cores}. It was shown earlier that over arbitrary structures, this
semantic notion for first-order logic corresponds to definability by
sentences. In this paper, we identify two properties of
classes of finite structures that ensure the above correspondence. The first is
based on well-quasi-ordering under the embedding relation. The second is a
logic-based combinatorial property that strictly generalizes the first. We show
that starting with classes satisfying any of these properties, the classes
obtained by applying operations like disjoint union, cartesian and tensor
products, or by forming words and trees over the classes, inherit the same
property. As a fallout, we obtain interesting classes of structures over which
an effective version of the {\L}o\'s-Tarski theorem holds.Comment: 28 pages, 1 figur
On infinite-finite duality pairs of directed graphs
The (A,D) duality pairs play crucial role in the theory of general relational
structures and in the Constraint Satisfaction Problem. The case where both
classes are finite is fully characterized. The case when both side are infinite
seems to be very complex. It is also known that no finite-infinite duality pair
is possible if we make the additional restriction that both classes are
antichains. In this paper (which is the first one of a series) we start the
detailed study of the infinite-finite case.
Here we concentrate on directed graphs. We prove some elementary properties
of the infinite-finite duality pairs, including lower and upper bounds on the
size of D, and show that the elements of A must be equivalent to forests if A
is an antichain. Then we construct instructive examples, where the elements of
A are paths or trees. Note that the existence of infinite-finite antichain
dualities was not previously known
Shared-memory Graph Truss Decomposition
We present PKT, a new shared-memory parallel algorithm and OpenMP
implementation for the truss decomposition of large sparse graphs. A k-truss is
a dense subgraph definition that can be considered a relaxation of a clique.
Truss decomposition refers to a partitioning of all the edges in the graph
based on their k-truss membership. The truss decomposition of a graph has many
applications. We show that our new approach PKT consistently outperforms other
truss decomposition approaches for a collection of large sparse graphs and on a
24-core shared-memory server. PKT is based on a recently proposed algorithm for
k-core decomposition.Comment: 10 pages, conference submissio
Parallel Graph Partitioning for Complex Networks
Processing large complex networks like social networks or web graphs has
recently attracted considerable interest. In order to do this in parallel, we
need to partition them into pieces of about equal size. Unfortunately, previous
parallel graph partitioners originally developed for more regular mesh-like
networks do not work well for these networks. This paper addresses this problem
by parallelizing and adapting the label propagation technique originally
developed for graph clustering. By introducing size constraints, label
propagation becomes applicable for both the coarsening and the refinement phase
of multilevel graph partitioning. We obtain very high quality by applying a
highly parallel evolutionary algorithm to the coarsened graph. The resulting
system is both more scalable and achieves higher quality than state-of-the-art
systems like ParMetis or PT-Scotch. For large complex networks the performance
differences are very big. For example, our algorithm can partition a web graph
with 3.3 billion edges in less than sixteen seconds using 512 cores of a high
performance cluster while producing a high quality partition -- none of the
competing systems can handle this graph on our system.Comment: Review article. Parallelization of our previous approach
arXiv:1402.328
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