## Abstract

We present new polynomial-time approximation schemes (PTAS) for

several basic minimum-cost multi-connectivity problems in geometrical graphs.

We focus on low connectivity requirements. Each of our schemes either signifi-

cantly improves the previously known upper time-bound or is the first PTAS for

the considered problem.

We provide a randomized approximation scheme for finding a biconnected graph

spanning a set of points in a multi-dimensional Euclidean space and having the

expected total cost within (1+") of the optimum. For any constant dimension and

", our scheme runs in time O(n log n). It can be turned into Las Vegas one without

affecting its asymptotic time complexity, and also efficiently derandomized.

The only previously known truly polynomial-time approximation (randomized)

scheme for this problem runs in expected time n (log n)O((log log n)9) in the

simplest planar case. The efficiency of our scheme relies on transformations of

nearly optimal low cost special spanners into sub-multigraphs having good decomposition

and approximation properties and a simple subgraph connectivity

characterization. By using merely the spanner transformations, we obtain a very

fast polynomial-time approximation scheme for finding a minimum-cost k-edge

connected multigraph spanning a set of points in a multi-dimensional Euclidean

space. For any constant dimension, ", and k, this PTAS runs in time O(n log n).

Furthermore, by showing a low-cost transformation of a k-edge connected graph

maintaining the k-edge connectivity and developing novel decomposition properties,

we derive a PTAS for Euclidean minimum-cost k-edge connectivity. It is

substantially faster than that previously known.

Finally, by extending our techniques, we obtain the first PTAS for the problem

of Euclidean minimum-cost Steiner biconnectivity. This scheme runs in time

O(n log n) for any constant dimension and ". As a byproduct, we get the first

known non-trivial upper bound on the number of Steiner points in an optimal

solution to an instance of Euclidean minimum-cost Steiner biconnectivity.

several basic minimum-cost multi-connectivity problems in geometrical graphs.

We focus on low connectivity requirements. Each of our schemes either signifi-

cantly improves the previously known upper time-bound or is the first PTAS for

the considered problem.

We provide a randomized approximation scheme for finding a biconnected graph

spanning a set of points in a multi-dimensional Euclidean space and having the

expected total cost within (1+") of the optimum. For any constant dimension and

", our scheme runs in time O(n log n). It can be turned into Las Vegas one without

affecting its asymptotic time complexity, and also efficiently derandomized.

The only previously known truly polynomial-time approximation (randomized)

scheme for this problem runs in expected time n (log n)O((log log n)9) in the

simplest planar case. The efficiency of our scheme relies on transformations of

nearly optimal low cost special spanners into sub-multigraphs having good decomposition

and approximation properties and a simple subgraph connectivity

characterization. By using merely the spanner transformations, we obtain a very

fast polynomial-time approximation scheme for finding a minimum-cost k-edge

connected multigraph spanning a set of points in a multi-dimensional Euclidean

space. For any constant dimension, ", and k, this PTAS runs in time O(n log n).

Furthermore, by showing a low-cost transformation of a k-edge connected graph

maintaining the k-edge connectivity and developing novel decomposition properties,

we derive a PTAS for Euclidean minimum-cost k-edge connectivity. It is

substantially faster than that previously known.

Finally, by extending our techniques, we obtain the first PTAS for the problem

of Euclidean minimum-cost Steiner biconnectivity. This scheme runs in time

O(n log n) for any constant dimension and ". As a byproduct, we get the first

known non-trivial upper bound on the number of Steiner points in an optimal

solution to an instance of Euclidean minimum-cost Steiner biconnectivity.

Original language | English |
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Title of host publication | Automata, languages and programming / Lecture notes in computer science |

Publisher | Springer |

Pages | 856-868 |

Volume | 1853 |

ISBN (Print) | 3540677151 |

Publication status | Published - 2000 |

Event | 27th international colloquium / ICALP 2000 - Geneva, Switzerland Duration: 2000 Jul 9 → 2000 Jul 15 |

### Publication series

Name | |
---|---|

Volume | 1853 |

### Conference

Conference | 27th international colloquium / ICALP 2000 |
---|---|

Country/Territory | Switzerland |

City | Geneva |

Period | 2000/07/09 → 2000/07/15 |

## Subject classification (UKÄ)

- Computer Science

## Keywords

- fast approximation schemes
- Euclidean multi-connectivity problems
- geometrical graphs