Approximation of minimum weight spanners for sparse graphs

  • Authors:

  • Feodor F. Dragan;Fedor V. Fomin;Petr A. Golovach

  • Affiliations:

  • Department of Computer Science, Kent State University, Kent, OH 44242, USA;Department of Informatics, University of Bergen, PB 7803, N-5020 Bergen, Norway;School of Engineering and Computing Sciences, Durham University, South Road, DH1 3LE Durham, UK

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2011

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Abstract

A t-spanner of a graph G is its spanning subgraph S such that the distance between every pair of vertices in S is at most t times their distance in G. The sparsestt-spanner problem asks to find, for a given graph G and an integer t, a t-spanner of G with the minimum number of edges. The problem is known to be NP-hard for all t=2, and, even more, it is NP-hard to approximate it with ratio O(logn) for every t=2. For t=5, the problem remains NP-hard for planar graphs and the approximability status of the problem on planar graphs was open. We resolve this open issue by showing that the sparsestt-spanner problem admits the efficient polynomial time approximation scheme (EPTAS) for every t=1. Our result holds for a much wider class of graphs, namely, the class of apex-minor-free graphs, which contains the classes of planar and bounded genus graphs. Moreover, it is possible to extend our results to weighted apex-minor free graphs, when the maximum edge weight is bounded by some constant.