Algorithmica

, Volume 73, Issue 1, pp 42–62

Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems

  • Karl Bringmann
  • Christian Engels
  • Bodo Manthey
  • B. V. Raghavendra Rao
Article

DOI: 10.1007/s00453-014-9901-9

Cite this article as:
Bringmann, K., Engels, C., Manthey, B. et al. Algorithmica (2015) 73: 42. doi:10.1007/s00453-014-9901-9
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Abstract

Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The distance between two nodes is then the length of a shortest path (with respect to the weights drawn) that connects these nodes. We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the \(k\)-median problem, as well as the running-time of the 2-opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worst-case bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons.

Keywords

Random shortest paths First passage percolation Approximation algorithms Random metrics 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Karl Bringmann
    • 1
  • Christian Engels
    • 2
  • Bodo Manthey
    • 3
  • B. V. Raghavendra Rao
    • 4
  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Saarland UniversitySaarbrückenGermany
  3. 3.University of TwenteEnschedeThe Netherlands
  4. 4.Indian Institute of Technology MadrasChennaiIndia

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