How to remove cycles in an unweighted directed graph, such that the number of edges is maximised?

问题

Let G be an unweighted directed graph containing cycles. I'm looking for an algorithm which finds/creates all acyclic graphs G', composed of all vertices in G and a subset of edges of G, just small enough to make G' acyclic.

More formal: The desired algorithm consumes G and creates a set of acyclic graphs S, where each graph G' in S satisfies following properties:

1. G' contains all vertices of G.
2. G' contains a subset of edges of G, such that G' is acyclic.
3. The number of edges of G' is maximised. Which means: There is no G'' satisfying properties 1 and 2, such that G'' contains more edges then G' and G'' is acyclic.

Background: The original graph G models a pairwise ordering between elements. This can't be exploited as an ordering over all elements due to cycles in the graph. The maximal acyclic graphs G' therefore should model a best-possible approximation to this ordering, trying to respect as much of the pairwise ordering relation as possible.

In a naive approach, one could remove all possible combinations of edges and check for acyclicity after each removal. In this case there is a strongly branching tree of variations meaning bad time and space complexity.

Note: The problem may be related to a spanning tree, and you could define the G' graphs as a kind of directed spanning tree. But keep in mind that in my scenario a pair of edges in G' may have the same starting or the same ending vertex. This conflicts with some definitions of directed spanning trees used in literature.

EDIT: Added intuitive description, background information and note related to spanning trees.

回答1:

This problem is called Feedback Arc Set. Since it is NP-hard, it is unlikely that you will find a scalable fast algorithm. However, if your instances are small, then algorithms such as the one from the paper “On enumerating all minimal solutions of feedback problems” by B. Schwikowski and E. Speckenmeyer might work.

回答2:

This may be helpful: An Algorithm for Enumerating all Directed Spanning Trees in a Directed Graph.

回答3:

If your goal is to remove cycle edges (breaking cycles) while preserving graph hierarchies (structures) as much as possible, this work maybe helpful: https://github.com/zhenv5/breaking_cycles_in_noisy_hierarchies

来源：https://stackoverflow.com/questions/6284469/how-to-remove-cycles-in-an-unweighted-directed-graph-such-that-the-number-of-ed