The arboricity of a graph is the minimum number of forests it can be partitioned into. Previous approximation schemes were nonconstructive, i.e., they only approximated the arboricity as a value without computing a corresponding forest partition, as they operate on the related pseudoforest partitions or the dual problem. We propose an algorithm for converting a partition of @math pseudoforests into a partition of @math forests in @math time, where @math is the inverse Ackermann function, when @math expected time is allowed for pre-computation of a perfect hash function. Without perfect hashing, we obtain @math with a data structure by Brodal and Fagerberg that stores graphs of arboricity @math . For every fixed @math , the latter result implies a constructive @math -approximation algorithm with runtime @math by using Kowalik's approximation scheme for pseudoforest partitions. Our algorithm might help in designing a faster exact arboricity algorithm. We also make several remarks on approximation algorithms for the pseudoarboricity and the equivalent graph orientations with smallest maximum indegree, and correct some mistakes made in the literature.
This paper studies the problem of orienting all edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be -hard generally. In this paper we first give optimal orientation algorithms which run in polynomial time for the following special cases: (i) the input is an unweighted graph, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, -approximation algorithm for the general case, where wmax and wmin are the maximum and the minimum weights of edges, respectively, and e is some small positive real number that depends on the input.