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On the Hardness of Constructing Minimal 2-Connected Spanning Subgraphs in Complete Graphs with Sharpened Triangle Inequality

Extended Abstract
  • Hans-Joachim Böckenhauer
  • Dirk Bongartz
  • Juraj Hromkovič
  • Ralf Klasing
  • Guido Proietti
  • Sebastian Seibert
  • Walter Unger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2556)

Abstract

In this paper we investigate the problem of finding a 2- connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, both for the edge- and for the vertex-connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval [1,1 + ε], for an arbitrary small ε > 0. This result implies the first explicit lower bound on the approximability of the general problems.

On the other hand, if the input graph satisfies the sharpened β-triangle inequality, then a (2/3 + 1/3 . β/1-β)-approximation algorithm is designed. This ratio tends to 1 with β tending to 1/2, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as β < 5/7.

Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened β-triangle inequality with β < 2/3, then this ratio can be improved to β/1-β.

Keywords

Approximation algorithm inapproximability minimum-cost biconnected spanning subgraph augmentation 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Dirk Bongartz
    • 1
  • Juraj Hromkovič
    • 1
  • Ralf Klasing
    • 2
  • Guido Proietti
    • 3
  • Sebastian Seibert
    • 1
  • Walter Unger
    • 1
  1. 1.Lehrstuhl für Informatik IAachenGermany
  2. 2.Department of Computer ScienceKing’s College London StrandLondonUK
  3. 3.Dipartimento di InformaticaUniversità di L’Aquila, 67010 L’Aquila, Italy and Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”RomaItaly

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