51,032 research outputs found
Finding Near-Optimal Independent Sets at Scale
The independent set problem is NP-hard and particularly difficult to solve in
large sparse graphs. In this work, we develop an advanced evolutionary
algorithm, which incorporates kernelization techniques to compute large
independent sets in huge sparse networks. A recent exact algorithm has shown
that large networks can be solved exactly by employing a branch-and-reduce
technique that recursively kernelizes the graph and performs branching.
However, one major drawback of their algorithm is that, for huge graphs,
branching still can take exponential time. To avoid this problem, we
recursively choose vertices that are likely to be in a large independent set
(using an evolutionary approach), then further kernelize the graph. We show
that identifying and removing vertices likely to be in large independent sets
opens up the reduction space---which not only speeds up the computation of
large independent sets drastically, but also enables us to compute high-quality
independent sets on much larger instances than previously reported in the
literature.Comment: 17 pages, 1 figure, 8 tables. arXiv admin note: text overlap with
arXiv:1502.0168
Fine-grained dichotomies for the Tutte plane and Boolean #CSP
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of
evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard
almost everywhere, and the remaining points admit polynomial-time algorithms.
Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination
with Curticapean [7], extended the #P-hardness results to tight lower bounds
under the counting exponential time hypothesis #ETH, with the exception of the
line , which was left open. We complete the dichotomy theorem for the
Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs
of a given -vertex graph cannot be determined in time unless
#ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann
[6] for counting the number of satisfying assignments to a constraint
satisfaction problem instance over the Boolean domain. We prove that all
#P-hard cases are also hard under #ETH. The main ingredient is to prove that
the number of independent sets in bipartite graphs with vertices cannot be
computed in time unless #ETH fails. In order to prove our results,
we use the block interpolation idea by Curticapean [7] and transfer it to
systems of linear equations that might not directly correspond to
interpolation.Comment: 16 pages, 1 figur
JGraphT -- A Java library for graph data structures and algorithms
Mathematical software and graph-theoretical algorithmic packages to
efficiently model, analyze and query graphs are crucial in an era where
large-scale spatial, societal and economic network data are abundantly
available. One such package is JGraphT, a programming library which contains
very efficient and generic graph data-structures along with a large collection
of state-of-the-art algorithms. The library is written in Java with stability,
interoperability and performance in mind. A distinctive feature of this library
is the ability to model vertices and edges as arbitrary objects, thereby
permitting natural representations of many common networks including
transportation, social and biological networks. Besides classic graph
algorithms such as shortest-paths and spanning-tree algorithms, the library
contains numerous advanced algorithms: graph and subgraph isomorphism; matching
and flow problems; approximation algorithms for NP-hard problems such as
independent set and TSP; and several more exotic algorithms such as Berge graph
detection. Due to its versatility and generic design, JGraphT is currently used
in large-scale commercial, non-commercial and academic research projects. In
this work we describe in detail the design and underlying structure of the
library, and discuss its most important features and algorithms. A
computational study is conducted to evaluate the performance of JGraphT versus
a number of similar libraries. Experiments on a large number of graphs over a
variety of popular algorithms show that JGraphT is highly competitive with
other established libraries such as NetworkX or the BGL.Comment: Major Revisio
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
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?
Correlations in the signal observed via functional Magnetic Resonance Imaging
(fMRI), are expected to reveal the interactions in the underlying neural
populations through hemodynamic response. In particular, they highlight
distributed set of mutually correlated regions that correspond to brain
networks related to different cognitive functions. Yet graph-theoretical
studies of neural connections give a different picture: that of a highly
integrated system with small-world properties: local clustering but with short
pathways across the complete structure. We examine the conditional independence
properties of the fMRI signal, i.e. its Markov structure, to find realistic
assumptions on the connectivity structure that are required to explain the
observed functional connectivity. In particular we seek a decomposition of the
Markov structure into segregated functional networks using decomposable graphs:
a set of strongly-connected and partially overlapping cliques. We introduce a
new method to efficiently extract such cliques on a large, strongly-connected
graph. We compare methods learning different graph structures from functional
connectivity by testing the goodness of fit of the model they learn on new
data. We find that summarizing the structure as strongly-connected networks can
give a good description only for very large and overlapping networks. These
results highlight that Markov models are good tools to identify the structure
of brain connectivity from fMRI signals, but for this purpose they must reflect
the small-world properties of the underlying neural systems
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