Algorithmica

, Volume 77, Issue 3, pp 686–713 | Cite as

Line-Distortion, Bandwidth and Path-Length of a Graph

  • Feodor F. Dragan
  • Ekkehard Köhler
  • Arne Leitert
Article

Abstract

For a graph \(G=(V,E)\) the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices xy the distance on the line \(|f(x) - f(y)|\) can be bounded by the term \(d_G(x, y)\le |f(x)-f(y)|\le k \, d_G(x, y)\), where \(d_G(x, y)\) is the distance in the graph. The minimum bandwidth problem minimizes the term \(\max _{uv\in E}|f(u)-f(v)|\), where f is a mapping of the vertices of G into the integers \(\{1, \ldots , n\}\). We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson–Seymour’s path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show:
  • there is a simple polynomial time algorithm that embeds an arbitrary unweighted input graph G into the line with distortion \(\mathcal{O}(k^2)\), where k is the minimum line-distortion of G;

  • if a graph G can be embedded into the line with distortion k, then G admits a Robertson–Seymour’s path-decomposition with bags of diameter at most k in G;

  • for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem;

  • there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph;

  • AT-free graphs and some intersection families of graphs have path-length at most 2;

  • for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.

Keywords

Graph algorithms Approximation algorithms Minimum line-distortion Minimum bandwidth Robertson–Seymour’s path-decomposition Path-length AT-free graphs 

Notes

Acknowledgments

We are very grateful to anonymous referees for many useful suggestions.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Feodor F. Dragan
    • 1
  • Ekkehard Köhler
    • 2
  • Arne Leitert
    • 1
  1. 1.Algorithmic Research Laboratory, Department of Computer ScienceKent State UniversityKentUSA
  2. 2.Mathematisches InstitutBrandenburgische Technische Universität CottbusCottbusGermany

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