Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics

Stefan Klootwijk*, Bodo Manthey

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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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.
Original languageEnglish
Title of host publication31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)
EditorsMichael Drmota, Clemens Heuberger
Place of PublicationDagstuhl
PublisherDagstuhl
Number of pages16
ISBN (Electronic)978-3-95977-147-4
DOIs
Publication statusPublished - 10 Jun 2020
Event31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2020 - Virtual event
Duration: 15 Jun 202019 Jun 2020
Conference number: 31

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Volume159
ISSN (Print)1868-8969

Conference

Conference31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2020
Abbreviated titleAofA 2020
Period15/06/2019/06/20

Keywords

  • Approximation algorithms
  • Firstpassage percolation
  • Random metrics
  • Random shortest paths

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