DATE: | Wednesday, March 15th |

TIME: | 4:30 PM - 5:30 PM |

PLACE: | McConnell 320 |

TITLE: | Approximability of the Minimum-Cost k-Connected Spanning Subgraph Problem |

SPEAKER: | Artur Czumaj, New Jersey Institute of Technology |

In the first part of the presentation I will describe polynomial-time approximation schemes (PTAS) for several basic minimum-cost multi-connectivity problems in geometrical graphs. I describe 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 $(1+\epsilon)$ of the optimum. For any constant dimension and $\epsilon$, the 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.

Further, I will briefly explain the extension of this approach to a PTAS for finding a minimum-cost k-edge connected graph/multigraph spanning a set of points in a multi-dimensional Euclidean space and to a PTAS for the problem of Euclidean minimum-cost Steiner biconnectivity.

In the second part of the talk I will discuss hardness of approximations for the minimum-cost k-vertex- and k-edge-connected spanning subgraph problems. The only inapproximability result known so far states that the k-edge-connectivity problem in unweighted graphs does not have a PTAS unless P=NP, even for k=2. I shall present a simpler proof of this result that holds even for graphs of bounded degree, and provide the first proof that finding a PTAS for the k-vertex-connectivity problem in unweighted graphs is NP-hard even for k=2 and for graphs of bounded degree.

This is a joint work Andrzej Lingas (Lund University, Sweden)

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