A primal-dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs.

*(English)*Zbl 0920.90137Summary: Recently, A. Becker and D. Geiger [‘Approximation algorithms for the loop cutset problem’, in: Proc. 10th Conf. on Uncertainty in Artificial Intelligence, 60-68 (1994)] and V. Bafna, P. Berman and T. Fujito [‘Constant ratio approximation of the weighted feedback vertex set problem for undirected graphs’, in: J. Staples, P. Eades, N. Katoh, A. Moffat (eds.), ISAAC ’95, Algorithms and Computation, Lect. Notes Computer Sci. 1004, 142-151 (1995)] gave 2-approximation algorithms for the feedback vertex set problem in undirected graphs. We show how their algorithms can be explained in terms of the primal-dual method for approximation algorithms, which has been used to derive approximation algorithms for network design problems. In the process, we give a new integer programming formulation for the feedback vertex set problem whose integrality gap is at worst a factor of two; the well-known cycle formulation has an integrality gap of \(\Theta(\log n)\), as shown by E. Even, J. Noor, B. Schieber and L. Zosin [‘Approximating minimum subset feedback sets in undirected graphs with applications’, in: Proc. 4th Israel Symp. on Theory of Computing and Systems, 78-88 (1996)]. We also give a new 2-approximation algorithm for the problem which is a simplification of the Bafna et al. algorithm.

##### MSC:

90C35 | Programming involving graphs or networks |

##### Keywords:

combinatorial optimization; 2-approximation algorithms; feedback vertex set problem; undirected graphs; network design
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\textit{F. A. Chudak} et al., Oper. Res. Lett. 22, No. 4--5, 111--118 (1998; Zbl 0920.90137)

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##### References:

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[8] | D.S. Hochbaum, Approximating covering and packing problems: Set cover, vertex cover, independent set, and related problems, in: D.S. Hochbaum (Ed.), Approximation Algorithms for NP-hard Problems, Ch. 3, PWS, Boston, 1996. |

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