The densest subgraph problem, which asks for a subgraph with the maximum edges-to-vertices ratio d * , is solvable in polynomial time. We discuss algorithms for this problem and the computation of a graph orientation with the lowest maximum indegree, which is equal to ⌈d * ⌉. This value also equals the pseudoarboricity of the graph. We show that it can be computed in O(|E| 3/2 √ log log d * ) time, and that better estimates can be given for graph classes where d * satisfies certain asymptotic bounds. These runtimes are achieved by accelerating a binary search with an approximation scheme, and a runtime analysis of Dinitz's algorithm on flow networks where all arcs, except the source and sink arcs, have unit capacity. We experimentally compare implementations of various algorithms for the densest subgraph and pseudoarboricity problems. In flow-based algorithms, Dinitz's algorithm performs significantly better than push-relabel algorithms on all instances tested.

The arboricity Γ of a graph is the minimum number of forests its edge set 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. This is because they operate on the related 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 when perfect hashing is used. When applied to a pseudoforest partition obtained from Kowalik's approximation scheme, our conversion implies a constructive (1 + )-approximation algorithm with runtime O(m log n log Γ −1 ) for every > 0. For fixed , the runtime can be reduced to O(m log n).Our conversion also implies a near-exact algorithm that computes a partition into at most Γ + 2 forests in O(m log n Γ log * Γ) time. It might also pave the way to faster exact arboricity algorithms.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. AcknowledgementsThe authors would like to thank Łukasz Kowalik for discussions on the matter, as well as Ernst Althaus for simplifying the algorithm that eliminates duplicate colors.We can employ a lemma by Duncan, Eppstein and Kobourov for a first result.Lemma 4 ([17]). Let G be a simple graph with ∆(G) ≤ 3. Then G can be partitioned into two linear forests in linear time.Theorem 5. A pseudoforest partition (P 1 , P 2 , P 3 ) can be converted into a partition of five forests, two of which are linear forests, in linear time.Proof. Partition each P i into a forest F i and a P i -matching M i according to Lemma 3 in linear time. Consider the graph on V with edges M 1 ∪ M 2 ∪ M 3 . Clearly, it has maximum degree three. Thus it can be partitioned into two linear forests by Lemma 4 in linear time.This implies a linear-time conversion of k pseudoforests into 5k/3 forests. In the following theorem, we give a better result by modifying an initially arbitrary choice of matchings. (91)90100-X.16 E. A. Dinic. Algorithm for solution of a problem of maximum flow in a network with power estimation (from Russian).

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