# 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} }

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

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- Mathematics, Computer ScienceSIAM J. Discret. Math.
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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.

### Parameterized algorithms for feedback set problems and their duals in tournaments

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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 ScienceMFCS
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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

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- 2008

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

- MathematicsTheory 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

- Mathematics, Computer Science
- 2011

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

- MathematicsAlgorithmica
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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 ScienceComb.
- 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.

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