138 research outputs found
Scalable Kernelization for Maximum Independent Sets
The most efficient algorithms for finding maximum independent sets in both
theory and practice use reduction rules to obtain a much smaller problem
instance called a kernel. The kernel can then be solved quickly using exact or
heuristic algorithms---or by repeatedly kernelizing recursively in the
branch-and-reduce paradigm. It is of critical importance for these algorithms
that kernelization is fast and returns a small kernel. Current algorithms are
either slow but produce a small kernel, or fast and give a large kernel. We
attempt to accomplish both of these goals simultaneously, by giving an
efficient parallel kernelization algorithm based on graph partitioning and
parallel bipartite maximum matching. We combine our parallelization techniques
with two techniques to accelerate kernelization further: dependency checking
that prunes reductions that cannot be applied, and reduction tracking that
allows us to stop kernelization when reductions become less fruitful. Our
algorithm produces kernels that are orders of magnitude smaller than the
fastest kernelization methods, while having a similar execution time.
Furthermore, our algorithm is able to compute kernels with size comparable to
the smallest known kernels, but up to two orders of magnitude faster than
previously possible. Finally, we show that our kernelization algorithm can be
used to accelerate existing state-of-the-art heuristic algorithms, allowing us
to find larger independent sets faster on large real-world networks and
synthetic instances.Comment: Extended versio
Improved FPT algorithms for weighted independent set in bull-free graphs
Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT
algorithm for Weighted Independent Set in bull-free graphs parameterized by the
weight of the solution, running in time . In this article
we improve this running time to . As a byproduct, we also
improve the previous Turing-kernel for this problem from to .
Furthermore, for the subclass of bull-free graphs without holes of length at
most for , we speed up the running time to . As grows, this running time is
asymptotically tight in terms of , since we prove that for each integer , Weighted Independent Set cannot be solved in time in the class of -free graphs unless the
ETH fails.Comment: 15 page
Galactic Token Sliding
International audienc
Symmetry in Graph Theory
This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view
Fast Parallel Algorithms on a Class of Graph Structures With Applications in Relational Databases and Computer Networks.
The quest for efficient parallel algorithms for graph related problems necessitates not only fast computational schemes but also requires insights into their inherent structures that lend themselves to elegant problem solving methods. Towards this objective efficient parallel algorithms on a class of hypergraphs called acyclic hypergraphs and directed hypergraphs are developed in this thesis. Acyclic hypergraphs are precisely chordal graphs and their subclasses, and they have applications in relational databases and computer networks. In this thesis, first, we present efficient parallel algorithms for the following problems on graphs. (1) determining whether a graph is strongly chordal, ptolemaic, or a block graph. If the graph is strongly chordal, determine the strongly perfect vertex elimination ordering. (2) determining the minimal set of edges needed to make an arbitrary graph strongly chordal, ptolemaic, or a block graph. (3) determining the minimum cardinality dominating set, connected dominating set, total dominating set, and the domatic number of a strongly chordal graph. Secondly, we show that the query implication problem (Q\sb1\ \to\ Q\sb2) on two queries, which is to determine whether the data retrieved by query Q\sb1 is always a subset of the data retrieved by query Q\sb2, is not even in NP and in fact complete in \Pi\sb2\sp{p}. We present several \u27fine-grain\u27 analyses of the query implication problem and show that the query implication can be solved in polynomial time given chordal queries. Thirdly, we develop efficient parallel algorithms for manipulating directed hypergraphs H such as finding a directed path in H, closure of H, and minimum equivalent hypergraph of H. We show that finding a directed path in a directed hypergraph is inherently sequential. For directed hypergraphs with fixed degree and diameter we present NC algorithms for manipulations. Directed hypergraphs are representation schemes for functional dependencies in relational databases. Finally, we also present an efficient parallel algorithm for multi-dimensional range search. We show that a set of points in a rectangular parallelepiped can be obtained in O(logn) time with only 2.log\sp2 n 10.logn + 14 processors on a EREW-PRAM. A nontrivial implementation technique on the hypercube parallel architecture is also presented. Our method can be easily generalized to the case of d-dimensional range search
Visualized Algorithm Engineering on Two Graph Partitioning Problems
Concepts of graph theory are frequently used by computer scientists as abstractions when modeling a problem. Partitioning a graph (or a network) into smaller parts is one of the fundamental algorithmic operations that plays a key role in classifying and clustering. Since the early 1970s, graph partitioning rapidly expanded for applications in wide areas. It applies in both engineering applications, as well as research. Current technology generates massive data (“Big Data”) from business interactions and social exchanges, so high-performance algorithms of partitioning graphs are a critical need.
This dissertation presents engineering models for two graph partitioning problems arising from completely different applications, computer networks and arithmetic. The design, analysis, implementation, optimization, and experimental evaluation of these models employ visualization in all aspects. Visualization indicates the performance of the implementation of each Algorithm Engineering work, and also helps to analyze and explore new algorithms to solve the problems. We term this research method as “Visualized Algorithm Engineering (VAE)” to emphasize the contribution of the visualizations in these works.
The techniques discussed here apply to a broad area of problems: computer networks, social networks, arithmetic, computer graphics and software engineering. Common terminologies accepted across these disciplines have been used in this dissertation to guarantee practitioners from all fields can understand the concepts we introduce
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