# Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

@article{Even1998ApproximatingMF,
title={Approximating Minimum Feedback Sets and Multicuts in Directed Graphs},
author={Guy Even and Joseph Naor and Baruch Schieber and Madhu Sudan},
journal={Algorithmica},
year={1998},
volume={20},
pages={151-174}
}
• Published 1 February 1998
• Computer Science, Mathematics
• Algorithmica
Abstract. This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems…
278 Citations

### Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications

• Mathematics, Computer Science
SIAM J. Discret. Math.
• 2000
A polynomial time algorithm is provided for approximating the subset feedback edge set problem that achieves an approximation factor of two and a bootstrapping technique is employed to achieve the O(\log \tau^*) factor, which is the value of the optimal fractional solution.

### Graph and Election Problems Parameterized by Feedback Set Numbers

This work investigates the parameterized complexity of three related graph modification problems, given a directed graph, a distinguished vertex, and a positive integer k, with respect to the parameters "treewidth", " size of a feedback vertex set" and "size of a Feedback arc set".

### Towards a Polynomial Kernel for Directed Feedback Vertex Set

• Mathematics, Computer Science
MFCS
• 2017
Two main contributions are provided: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

### Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs

A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas

### Exact Localisations of Feedback Sets

• M. Hecht
• Mathematics
Theory of Computing Systems
• 2017
The notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in 𝓞(|V||E|3)$\mathcal {O}(| V||E |^{3})$ is introduced and weighted versions of the FASP and FVSP possess a Bellman decomposition.

### Subset feedback vertex set is fixed-parameter tractable

This paper shows that the SUBSET-FVS problem is fixed-parameter tractable when parametrized byjSj, and presents an algorithm which reduces the given instance to 2 k n O(1) instances with the size of S bounded by O(k 3 ), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai’s theorem.

### Algorithms and Kernels for Feedback Set Problems in Generalizations of Tournaments

• Mathematics
Algorithmica
• 2015
This paper gives polynomial time algorithms on several digraph classes that given an instance (D, k) of the problem returns an equivalent instance(D',k') such that the size of D and k is at most kO(1), and designs a subexponential algorithm for k-FAS running in time.

### Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs

• Mathematics, Computer Science
Comb.
• 1998
This work gives a ]-approxlmation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties, and uses the primaldual method for approximation algorithms as given in Goemans and Williamson.

## References

SHOWING 1-10 OF 34 REFERENCES

### Primal-Dual Approximation Algorithms for Feedback Problems

• Mathematics, Computer Science
IPCO
• 1996
A 9/4-approximation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties, using the primaldual method for approximation algorithms as given in Goemans and Williamson.

### Approximation Algorithms for Steiner and Directed Multicuts

• Computer Science, Mathematics
J. Algorithms
• 1997
AnO(log3(kt) approximation algorithm for the Steiner multicut problem, where k is the number of sets andtis the maximum cardinality of a set, improves theO(tlogk) bound that easily follows from the previously known multicut results.

### A Minimax Theorem for Directed Graphs

• Mathematics
• 1978
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

### An approximate max-flow min-cut relation for undirected multicommodity flow, with applications

• Computer Science
Comb.
• 1995
It is shown that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is thesum of demands.

### An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms

• Computer Science
[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
• 1988
The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Theta (log n) factor.

### Divide-and-conquer approximation algorithms via spreading metrics

• Computer Science, Mathematics
Proceedings of IEEE 36th Annual Foundations of Computer Science
• 1995
A polynomial time approximation algorithm for problems modelled by the novel divide-and-conquer paradigm, whose approximation factor is O (mi.tau), is presented.

### Approximate max-flow min-(multi)cut theorems and their applications

• Computer Science
STOC '93
• 1993
The proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of Leighton and Rao and Klein et al. and thereby obtain an improved bound for the latter problem.

### Approximate Max--ow Min-(multi)cut Theorems and Their Applications

• Computer Science
• 1993
The proof technique provides a uniied framework in which one can also analyse the case of ows with speciied demands, of Leighton-Rao and Klein et.al.

### Packing directed circuits fractionally

There is a set ofO (k logk log k logk) vertices meeting all directed circuits ofG, such that no “fractional” packing of directed circuit ofG has value >k, when every vertex is given “capacity” 1.