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Exact Minimum Weight Spanners via Column Generation

Authors Fritz Bökler , Markus Chimani , Henning Jasper , Mirko H. Wagner

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Fritz Bökler
  • Institute of Computer Science, Osnabrück University, Germany
Markus Chimani
  • Institute of Computer Science, Osnabrück University, Germany
Henning Jasper
  • Institute of Computer Science, Osnabrück University, Germany
Mirko H. Wagner
  • Institute of Computer Science, Osnabrück University, Germany

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Fritz Bökler, Markus Chimani, Henning Jasper, and Mirko H. Wagner. Exact Minimum Weight Spanners via Column Generation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.30

Abstract

Given a weighted graph G, a minimum weight α-spanner is a least-weight subgraph H ⊆ G that preserves minimum distances between all node pairs up to a factor of α. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [Markus Chimani and Finn Stutzenstein, 2022]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [Sigurd and Zachariasen, 2004] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [Ahmed et al., 2019], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable. In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16 000 nodes within 13 min. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Mathematical optimization
  • Theory of computation → Network optimization
Keywords
  • Graph spanners
  • ILP
  • algorithm engineering
  • experimental study

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