## Abstract

A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest 1994 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP-hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity, our algorithm guarantees a solution that is no more than 3/2 times the optimal. For 2-vertex connectivity, our algorithm guarantees a solution that is no more than 5/3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP-hard as well. We also consider the case where the graph has edge weights. For this case, we show that an approximation factor of 2 is possible in polynomial time for finding a k-edge connected spanning subgraph. This improves an approximation factor of 3 for k = 2, due to Frederickson and Ja´Ja´ [1981], and extends it for any k (with an increased running time though).

Original language | English (US) |
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Pages (from-to) | 214-235 |

Number of pages | 22 |

Journal | Journal of the ACM (JACM) |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Jan 3 1994 |

## Keywords

- Biconnectivity
- connectivity
- sparse subgraphs

## ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence