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

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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.

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Notes

  1. Exponential distributions are technically the easiest to handle because they are memoryless. We will discuss other distributions in Sect. 6.

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Authors and Affiliations

  1. Max Planck Institute for Informatics, Saarbrücken, Germany

    Karl Bringmann

  2. Saarland University, Saarbrücken, Germany

    Christian Engels

  3. University of Twente, Enschede, The Netherlands

    Bodo Manthey

  4. Indian Institute of Technology Madras, Chennai, India

    B. V. Raghavendra Rao

Authors
  1. Karl Bringmann
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  2. Christian Engels
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  3. Bodo Manthey
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  4. B. V. Raghavendra Rao
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Corresponding author

Correspondence to Bodo Manthey.

Additional information

Karl Bringmann is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this research is supported in part by this Google Fellowship.

An extended abstract of this work has appeared in the Proceedings of the 38th Int. Symp. on Mathematical Foundations of Computer Science (MFCS 2013).

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Bringmann, K., Engels, C., Manthey, B. et al. Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems. Algorithmica 73, 42–62 (2015). https://doi.org/10.1007/s00453-014-9901-9

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  • DOI: https://doi.org/10.1007/s00453-014-9901-9

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