A Constructive Arboricity Approximation Scheme

Abstract

The arboricity 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 forests in 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-approximation algorithm for the arboricity with runtime for every. For fixed, the runtime can be reduced to. Moreover, our conversion implies a near-exact algorithm that computes a partition into at most forests in time. It might also pave the way to faster exact arboricity algorithms.