## Abstract

Let G = (V, E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ⊂ V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a Δ-approximation algorithm for the subset feedback vertex set problem, where Δ is the maximum degree in G. We also consider the multicut problem and show how to achieve an O(log τ*) approximation factor for this problem, where τ* is the value of the optimal fractional solution. To achieve the O(log τ*) factor we employ a bootstrapping technique.

Original language | English |
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Pages (from-to) | 255-267 |

Number of pages | 13 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2000 |

## Keywords

- Approximation algorithms
- Combinatorial optimization
- Feedback edge set
- Feedback vertex set
- Multicut
- Subset feedback set