5,511 research outputs found
On the complexity of color-avoiding site and bond percolation
The mathematical analysis of robustness and error-tolerance of complex
networks has been in the center of research interest. On the other hand, little
work has been done when the attack-tolerance of the vertices or edges are not
independent but certain classes of vertices or edges share a mutual
vulnerability. In this study, we consider a graph and we assign colors to the
vertices or edges, where the color-classes correspond to the shared
vulnerabilities. An important problem is to find robustly connected vertex
sets: nodes that remain connected to each other by paths providing any type of
error (i.e. erasing any vertices or edges of the given color). This is also
known as color-avoiding percolation. In this paper, we study various possible
modeling approaches of shared vulnerabilities, we analyze the computational
complexity of finding the robustly (color-avoiding) connected components. We
find that the presented approaches differ significantly regarding their
complexity.Comment: 14 page
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
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