Michail
Dimitrios
exponent
problem
time
vector
time O
new approximation algorithm
complete Euclidean graph
surface reconstruction
extension
space
matrix multiplication
minimum cycle basis
number
true
new algorithm
cycle basis
structural engineering
2007-01-01
multiplication
sum
https://doi.org/10.1007/978-3-540-70918-3_44
2022-12-01T06:51
planar graphs
edge-weighted graph G
low weight
cycle space
A set
chemistry
n vertices
context
https://scigraph.springernature.com/explorer/license/
We consider the problem of computing an approximate minimum cycle basis of an undirected edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}_2$\end{document} generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction.We present two new algorithms to compute an approximate minimum cycle basis. For any integer k ≥ 1, we give (2k − 1)-approximation algorithms with expected running time O(kmn1 + 2/k + mn(1 + 1/k)(ω − 1)) and deterministic running time O( n3 + 2/k ), respectively. Here ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. Both algorithms are o( mω) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(mω) bound.We also present a 2-approximation algorithm with \documentclass[12pt]{minimal}
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\begin{document}$O(m^{\omega}\sqrt{n\log n})$\end{document} expected running time, a linear time 2-approximation algorithm for planar graphs and an O(n3) time 2.42-approximation algorithm for the complete Euclidean graph in the plane.
weight
2007-01-01
vertices
best exponent
reconstruction
drop
set
Euclidean graph
analysis
basis
edge
network
512-523
chapters
approximation algorithm
New Approximation Algorithms for Minimum Cycle Bases of Graphs
cycle
first time
linear time
vector space
number of contexts
engineering
dense graphs
electrical network
plane
graph
graph G
chapter
incidence vectors
algorithm
Mehlhorn
Kurt
Indian Institute of Science, Bangalore, India
Indian Institute of Science, Bangalore, India
Telikepalli
Kavitha
Computation Theory and Mathematics
Pascal
Weil
Springer Nature - SN SciGraph project
pub.1035485296
dimensions_id
Wolfgang
Thomas
Max-Planck-Institut für Informatik, Saarbrücken, Germany
Max-Planck-Institut für Informatik, Saarbrücken, Germany
Information and Computing Sciences
STACS 2007
978-3-540-70917-6
978-3-540-70918-3
10.1007/978-3-540-70918-3_44
doi
Springer Nature