Exact Localisations of Feedback Sets

@article{Hecht2017ExactLO,
  title={Exact Localisations of Feedback Sets},
  author={Michael Hecht},
  journal={Theory of Computing Systems},
  year={2017},
  volume={62},
  pages={1048-1084}
}
  • M. Hecht
  • Published 24 February 2017
  • Mathematics
  • Theory of Computing Systems
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph G = (V, E) into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs Gel(e), Gsi(e) of all elementary cycles or simple cycles running through some arc e ∈ E, can be computed in π“ž|E|2$\mathcal {O}\big (|E|^{2}\big )$ and π“ž(|E|4… 

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References

SHOWING 1-10 OF 41 REFERENCES

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

TLDR
A combinatorial algorithm that computes a (1+Ι›) approximation to the fractional optimal feedback vertex set, and a generalization of these problems, in which the feedback set has to intersect only a subset of the directed cycles in the graph.

Approximation alogorithms for the maximum acyclic subgraph problem

TLDR
It is found that all graphs without two-cycles contain large acyclic subgraphs, a fact which was not previously known.

Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs

TLDR
It is proved that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Euleria subgraph with minimum degree at least m2/24n3, and the conjecture is motivated by a conjecture of BollobΓ‘s and Scott.

Finding a Minimum Feedback Arc Set in Reducible Flow Graphs

Computing Minimum Directed Feedback Vertex Set in O(1.9977n)

TLDR
An algorithm which, given a directed graph G, finds the minimum directed feedback vertex set (FVS) of G in Oβˆ—(1.9977n) time and polynomial space is proposed, believed to be the first algorithm computing the minimumdirected FVS faster than in O(2n).

A fixed-parameter algorithm for the directed feedback vertex set problem

The (parameterized) FEEDBACK VERTEX SET problem on directed graphs (i.e., the DFVS problem) is defined as follows: given a directed graph <i>G</i> and a parameter <i>k</i>, either construct a…

Digraphs - theory, algorithms and applications

TLDR
Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science, and it will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

A Minimax Theorem for Directed Graphs

This minimax equality was conjectured about a decade ago by one of the authors ([7; page 43], [8], [9]) and, independently, by Neil Robertson. It arose in the study of a problem posed several years…

On the acyclic subgraph polytope

TLDR
This work determines several classes of facets for the associated acyclic subgraph polytope and shows that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time.

Finding All the Elementary Circuits of a Directed Graph

An algorithm is presented which finds all the elementary circuits of a directed graph in time bounded by $O((n + e)(c + 1))$ and space bounded by $O(n + e)$, where there are n vertices, e edges and c…