On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality

  • Authors:

  • Hans-Joachim Böckenhauer;Dirk Bongartz;Juraj Hromkovič;Ralf Klasing;Guido Proietti;Sebastian Seibert;Walter Unger

  • Affiliations:

  • Lehrstuhl für Informatik I, RWTH Aachen, 52074 Aachen, Germany;Lehrstuhl für Informatik I, RWTH Aachen, 52074 Aachen, Germany;Lehrstuhl für Informatik I, RWTH Aachen, 52074 Aachen, Germany;MASCOTTE project, 13S-CNRS/INRIA/Université de Nice-Sophia Antipolis, 2004 Route des Lucioles, BP 93 F-06902 Sophia Antipolis Cedex, France;Dipartimento di Informatica, Università di L'Aquila, Via Vetoio, 67010 L'Aquila, Italy and Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti", CNR, Roma, Italy;Lehrstuhl für Informatik I, RWTH Aachen, 52074 Aachen, Germany;Lehrstuhl für Informatik I, RWTH Aachen, 52074 Aachen, Germany

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2004

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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, for both the edge and 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 version (i.e., for arbitrary graphs) of the problem. On the other hand, if the input graph satisfies the sharpened β-triangle inequality, then a (2/3 + 1/3 ċ β/1-β)-approximation algorithm is designed. This ratio tends to 1 with β tending to ½, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as β 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