A Constructive Arboricity Approximation Scheme

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.