6,074 research outputs found
Edge-partitioning graphs into regular and locally irregular components
International audienceA graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph , there exists a minimum integer such that admits an edge-partition into classes, each of which induces a locally irregular graph. In particular, they conjectured that for every , unless belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on is known for bipartite, and (2) no satisfactory general upper bound on is known. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most subgraphs. This result implies that every graph admits a decomposition into at most subgraphs whose components are regular or locally irregular
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
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
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph
partitioning can all be expressed in terms of graph Laplacian matrices. There
are a variety of practical approaches to solving these problems in serial.
However, as problem sizes increase and single core speeds stagnate, parallelism
is essential to solve such problems quickly. We present an unsmoothed
aggregation multigrid method for solving graph Laplacians in a distributed
memory setting. We introduce new parallel aggregation and low degree
elimination algorithms targeted specifically at irregular degree graphs. These
algorithms are expressed in terms of sparse matrix-vector products using
generalized sum and product operations. This formulation is amenable to linear
algebra using arbitrary distributions and allows us to operate on a 2D sparse
matrix distribution, which is necessary for parallel scalability. Our solver
outperforms the natural parallel extension of the current state of the art in
an algorithmic comparison. We demonstrate scalability to 576 processes and
graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm
Distributed Processing of Generalized Graph-Pattern Queries in SPARQL 1.1
We propose an efficient and scalable architecture for processing generalized
graph-pattern queries as they are specified by the current W3C recommendation
of the SPARQL 1.1 "Query Language" component. Specifically, the class of
queries we consider consists of sets of SPARQL triple patterns with labeled
property paths. From a relational perspective, this class resolves to
conjunctive queries of relational joins with additional graph-reachability
predicates. For the scalable, i.e., distributed, processing of this kind of
queries over very large RDF collections, we develop a suitable partitioning and
indexing scheme, which allows us to shard the RDF triples over an entire
cluster of compute nodes and to process an incoming SPARQL query over all of
the relevant graph partitions (and thus compute nodes) in parallel. Unlike most
prior works in this field, we specifically aim at the unified optimization and
distributed processing of queries consisting of both relational joins and
graph-reachability predicates. All communication among the compute nodes is
established via a proprietary, asynchronous communication protocol based on the
Message Passing Interface
Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids
This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Distributed-Memory Breadth-First Search on Massive Graphs
This chapter studies the problem of traversing large graphs using the
breadth-first search order on distributed-memory supercomputers. We consider
both the traditional level-synchronous top-down algorithm as well as the
recently discovered direction optimizing algorithm. We analyze the performance
and scalability trade-offs in using different local data structures such as CSR
and DCSC, enabling in-node multithreading, and graph decompositions such as 1D
and 2D decomposition.Comment: arXiv admin note: text overlap with arXiv:1104.451
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