## Abstract

A k-partite graph is a graph G = (V_{1},...,V_{k}, E), where V_{1},...,V_{k} are k non-empty disjoint independent sets of vertices. Such a graph is called complete k-partite if E = U_{i ≠ j} V_{i} × V_{j}. We discuss three variants of the following optimization problem: given a graph and a non-negative weight function on the vertices and edges, find a minimum weight set of vertices and incident edges whose removal from the graph leaves a complete k-partite graph. All the problems we consider are at least as hard to approximate as the weighted vertex cover problem. We use the local-ratio technique to develop 2-approximation algorithms for the first two variants of the problem. In particular, we present the first (linear time) 2-approximation algorithm for the edge clique complement problem. For other previously studied special cases our 2-approximation algorithms are better in terms of time complexity than the existing 2-approximation algorithms. We use approximation preserving reductions in order to (4 - 4/k)-approximate the third variant of the problem.

Original language | English |
---|---|

Pages (from-to) | 20-40 |

Number of pages | 21 |

Journal | Journal of Algorithms |

Volume | 42 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2002 |

Externally published | Yes |

### Bibliographical note

Funding Information:1 This research was supported by the Fund for the Promotion of Research at the Technion.

## Keywords

- Approximation algorithm
- Complete k-partite
- Edge deletion
- Local-ratio technique
- Vertex deletion