A PTAS for the Sparsest Spanners Problem on Apex-Minor-Free Graphs

  • Authors:

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

  • Affiliations:

  • Department of Computer Science, Kent State University, Kent, USA 44242;Department of Informatics, University of Bergen, Bergen, Norway 5020;Department of Informatics, University of Bergen, Bergen, Norway 5020

  • Venue:
  • MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

A t-spannerof a graph Gis a spanning subgraph Sin which the distance between every pair of vertices is at most ttimes their distance in G. The sparsestt-spannerproblem asks to find, for a given graph Gand an integer t, a t-spanner of Gwith the minimum number of edges. On general n-vertex graphs, the problem is known to be NP-hard for all t茂戮驴 2, and, even more, it is NP-hard to approximate it with ratio O(logn) for every t茂戮驴 2. For t茂戮驴 5, the problem remains NP-hard for planar graphs, and up to now the approximability status of the problem on planar graphs considered to be open. In this note, we resolve this open issue by showing that the sparsestt-spannerproblem admits a polynomial time approximation scheme (PTAS)for every t茂戮驴 1. Actually, our results hold for a much wider class of graphs, namely, on the class of apex-minor-free graphswhich contains the classes of planar and bounded genus graphs.