A Constructive Arboricity Approximation Scheme

The arboricity Gamma 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 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 O(in log n log Gamma epsilon(-1)) for every epsilon > 0. For fixed epsilon, the runtime can be reduced to O(m log n).Moreover, our conversion implies a near-exact algorithm that computes a partition into at most Gamma + 2 forests in O(m log n Gamma log* Gamma) time. It might also pave the way to faster exact arboricity algorithms.