Document Open Access Logo

Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics

Authors Stefan Klootwijk, Bodo Manthey

PDF
Thumbnail PDF

File

LIPIcs.AofA.2020.19.pdf
  • Filesize: 0.49 MB
  • 16 pages

Document Identifiers

Author Details

Stefan Klootwijk
  • Department of Applied Mathematics, University of Twente, Enschede, The Netherlands
Bodo Manthey
  • Department of Applied Mathematics, University of Twente, Enschede, The Netherlands

Funding

This research was supported by NWO grant 613.001.402.

Cite AsGet BibTex

Stefan Klootwijk and Bodo Manthey. Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.AofA.2020.19

Abstract

Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős - Rényi random graphs) have been used so far. In this paper we extend these findings to sparse graphs, with a focus on grid graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a grid graph, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Random network models
Keywords
  • Random shortest paths
  • Random metrics
  • Approximation algorithms
  • First-passage percolation

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Antonio Auffinger, Michael Damron, and Jack Hanson. 50 years of first passage percolation. arXiv e-prints, 2015. URL: http://arxiv.org/abs/1511.03262.
  2. David Avis, Burgess Davis, and John Michael Steele. Probabilistic analysis of a greedy heuristic for Euclidean matching. Probability in the Engineering and Informational Sciences, 2(2):143-156, 1988. URL: https://doi.org/10.1017/S0269964800000711.
  3. Jon Louis Bentley and James Benjamin Saxe. An analysis of two heuristics for the Euclidean traveling salesman problem. In Proceedings of the Eighteenth Annual Allerton Conference on Communication, Control, and Computing, pages 41-49, 1980. Google Scholar
  4. Béla Bollobás and Imre Leader. Edge-isoperimetric inequalities in the grid. Combinatorica, 11(4):299-314, 1991. URL: https://doi.org/10.1007/BF01275667.
  5. Jean-Louis Bon and Eugen Păltănea. Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis, 5(2):185-192, 1999. URL: https://doi.org/10.1023/A:1009605613222.
  6. Karl Bringmann, Christian Engels, Bodo Manthey, and B. V. Raghavendra Rao. Random shortest paths: Non-Euclidean instances for metric optimization problems. Algorithmica, 73(1):42-62, 2015. URL: https://doi.org/10.1007/s00453-014-9901-9.
  7. Barun Chandra, Howard Karloff, and Craig Tovey. New results on the old k-opt algorithm for the traveling salesman problem. SIAM Journal on Computing, 28(6):1998-2029, 1999. URL: https://doi.org/10.1137/S0097539793251244.
  8. Robert Davis and Armand Prieditis. The expected length of a shortest path. Information Processing Letters, 46(3):135-141, 1993. URL: https://doi.org/10.1016/0020-0190(93)90059-I.
  9. Christian Engels and Bodo Manthey. Average-case approximation ratio of the 2-opt algorithm for the TSP. Operations Research Letters, 37(2):83-84, 2009. URL: https://doi.org/10.1016/j.orl.2008.12.002.
  10. Matthias Englert, Heiko Röglin, and Berthold Vöcking. Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. Algorithmica, 68(1):190-264, 2014. URL: https://doi.org/10.1007/s00453-013-9801-4.
  11. Alan M. Frieze and Joseph E. Yukich. Probabilistic analysis of the TSP. In Gregory Gutin and Abraham P. Punnen, editors, The Traveling Salesman Problem and Its Variations, chapter 7, pages 257-307. Springer, Boston, MA, 2007. URL: https://doi.org/10.1007/0-306-48213-4_7.
  12. John Michael Hammersley and Dominic James Anthony Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Jerzy Neyman and Lucien Marie Le Cam, editors, Bernoulli 1713 Bayes 1763 Laplace 1813, Anniversary Volume, Proceedings of an International Research Seminar Statistical Laboratory, University of California, Berkeley 1963, pages 61-110. Springer Berlin Heidelberg, 1965. URL: https://doi.org/10.1007/978-3-642-49750-6_7.
  13. Refael Hassin and Eitan Zemel. On shortest paths in graphs with random weights. Mathematics of Operations Research, 10(4):557-564, 1985. URL: https://doi.org/10.1287/moor.10.4.557.
  14. C. Douglas Howard. Models of first-passage percolation. In Harry Kesten, editor, Probability on Discrete Structures, pages 125-173. Springer Berlin Heidelberg, 2004. URL: https://doi.org/10.1007/978-3-662-09444-0_3.
  15. Svante Janson. One, two and three times log n/n for paths in a complete graph with random weights. Combinatorics, Probability and Computing, 8(4):347-361, 1999. URL: https://doi.org/10.1017/S0963548399003892.
  16. Svante Janson. Tail bounds for sums of geometric and exponential variables. Statistics & Probability Letters, 135:1-6, 2018. URL: https://doi.org/10.1016/j.spl.2017.11.017.
  17. Richard Manning Karp and John Michael Steele. Probabilistic analysis of heuristics. In Eugene Leighton Lawler, Jan Karel Lenstra, Alexander Hendrik George Rinnooy Kan, and David Bernard Shmoys, editors, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, pages 181-205. John Wiley & Sons Ltd., 1985. Google Scholar
  18. Stefan Klootwijk, Bodo Manthey, and Sander K. Visser. Probabilistic analysis of optimization problems on generalized random shortest path metrics. In Gautam K. Das, Partha S. Mandal, Krishnendu Mukhopadhyaya, and Shin-ichi Nakano, editors, WALCOM: Algorithms and Computation, 13th International Conference, WALCOM 2019, Guwahati, India, February 27 - March 2, 2019, Proceedings, pages 108-120. Springer Nature Switzerland AG, 2019. URL: https://doi.org/10.1007/978-3-030-10564-8_9.
  19. Edward M. Reingold and Robert Endre Tarjan. On a greedy heuristic for complete matching. SIAM Journal on Computing, 10(4):676-681, 1981. URL: https://doi.org/10.1137/0210050.
  20. Daniel Richardson. Random growth in a tessellation. Mathematical Proceedings of the Cambridge Philosophical Society, 74(3):515-528, 1973. URL: https://doi.org/10.1017/S0305004100077288.
  21. Daniel J. Rosenkrantz, Richard Edwin Stearns, and Philip M. Lewis II. An analysis of several heuristics for the traveling salesman problem. SIAM Journal on Computing, 6(3):563-581, 1977. URL: https://doi.org/10.1137/0206041.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact