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Improved Approximation for the Directed Spanner Problem

  • Piotr Berman
  • Arnab Bhattacharyya
  • Konstantin Makarychev
  • Sofya Raskhodnikova
  • Grigory Yaroslavtsev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We give an \(O(\sqrt{n}\log n)\)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V,E) with nonnegative edge lengths dE → ℝ ≥ 0 and a stretch k ≥ 1, a subgraph H = (V,E H ) is a k-spanner of G if for every edge (u,v) ∈ E, the graph H contains a path from u to v of length at most k ·d(u,v). The previous best approximation ratio was \(\tilde{O}(n^{2/3})\), due to Dinitz and Krauthgamer (STOC ’11).

We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is \(\tilde{O}(n^{1/3})\) which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer, had approximation ratio \(\tilde{O}(\sqrt{n})\).

Keywords

Approximation Ratio Linear Program Relaxation Improve Approximation Edge Cost Short Path Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Piotr Berman
    • 1
  • Arnab Bhattacharyya
    • 2
  • Konstantin Makarychev
    • 3
  • Sofya Raskhodnikova
    • 1
  • Grigory Yaroslavtsev
    • 1
  1. 1.Pennsylvania State UniversityUSA
  2. 2.Massachusetts Institute of TechnologyUSA
  3. 3.IBM T.J. Watson Research CenterUSA

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