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A Constructive Arboricity Approximation Scheme

  • Markus BlumenstockEmail author
  • Frank Fischer
Conference paper
  • 221 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)

Abstract

The arboricity \(\varGamma \) of a graph is the minimum number of forests its edge set can be partitioned into. Previous approximation schemes were nonconstructive, i.e., they approximate the arboricity as a value without computing a corresponding forest partition. This is because they operate on pseudoforest partitions or the dual problem of finding dense subgraphs.

We propose an algorithm for converting a partition of k pseudoforests into a partition of \(k+1\) forests in \(\mathcal {O}(mk\log k + m \log n)\) time with a data structure by Brodal and Fagerberg that stores graphs of arboricity k. A slightly better bound can be given if perfect hashing is used. When applied to a pseudoforest partition obtained from Kowalik’s approximation scheme, our conversion implies a constructive \((1+\epsilon )\)-approximation algorithm for the arboricity with runtime \(\mathcal {O}(m \log n \log \varGamma \, \epsilon ^{-1})\) for every \(\epsilon > 0\). For fixed \(\epsilon \), the runtime can be reduced to \(\mathcal {O}(m \log n)\).

Moreover, our conversion implies a near-exact algorithm that computes a partition into at most \(\varGamma +2\) forests in \(\mathcal {O}(m\log n \,\varGamma \log ^* \varGamma )\) time. It might also pave the way to faster exact arboricity algorithms.

Keywords

Approximation algorithms Matroid partitioning 

Notes

Acknowledgements

The authors thank Łukasz Kowalik for discussions and Ernst Althaus for simplifying the algorithm that eliminates duplicate colors.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Computer ScienceJohannes Gutenberg University MainzMainzGermany

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