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Using Light Spanning Graphs for Passenger Assignment in Public Transport

Authors Irene Heinrich , Olli Herrala , Philine Schiewe , Topias Terho



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Author Details

Irene Heinrich
  • Department of Mathematics, TU Darmstadt, Germany
Olli Herrala
  • Systems Analysis Laboratory, Aalto University, Espoo, Finland
Philine Schiewe
  • Systems Analysis Laboratory, Aalto University, Espoo, Finland
Topias Terho
  • Systems Analysis Laboratory, Aalto University, Espoo, Finland

Acknowledgements

This work was developed during a guest stay of the first author at the Aalto University in Espoo, Finland.

Cite AsGet BibTex

Irene Heinrich, Olli Herrala, Philine Schiewe, and Topias Terho. Using Light Spanning Graphs for Passenger Assignment in Public Transport. In 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023). Open Access Series in Informatics (OASIcs), Volume 115, pp. 2:1-2:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/OASIcs.ATMOS.2023.2

Abstract

In a public transport network a passenger’s preferred route from a point x to another point y is usually the shortest path from x to y. However, it is simply impossible to provide all the shortest paths of a network via public transport. Hence, it is a natural question how a lighter sub-network should be designed in order to satisfy both the operator as well as the passengers. We provide a detailed analysis of the interplay of the following three quality measures of lighter public transport networks: - building cost: the sum of the costs of all edges remaining in the lighter network, - routing costs: the sum of all shortest paths costs weighted by the demands, - fairness: compared to the original network, for each two points the shortest path in the new network should cost at most a given multiple of the shortest path in the original network. We study the problem by generalizing the concepts of optimum communication spanning trees (Hu, 1974) and optimum requirement graphs (Wu, Chao, and Tang, 2002) to generalized optimum requirement graphs (GORGs), which are graphs achieving the social optimum amongst all subgraphs satisfying a given upper bound on the building cost. We prove that the corresponding decision problem is NP-complete, even on orb-webs, a variant of grids which serves as an important model of cities with a center. For the case that the given network is a parametric city (cf. Fielbaum et. al., 2017) with a heavy vertex we provide a polynomial-time algorithm solving the GORG-problem. Concerning the fairness-aspect, we prove that light spanners are a strong concept for public transport optimization. We underpin our theoretical considerations with integer programming-based experiments that allow us to compare the fairness-approach with the routing cost-approach as well as passenger assignment approaches from the literature.

Subject Classification

ACM Subject Classification
  • Applied computing → Transportation
  • Mathematics of computing → Discrete optimization
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Discrete optimization
  • Theory of computation → Design and analysis of algorithms
Keywords
  • passenger assignment
  • line planning
  • public transport
  • discrete optimization
  • complexity
  • algorithm design

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References

  1. Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discret. Comput. Geom., 9:81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  2. Júlia Baligács, Yann Disser, Irene Heinrich, and Pascal Schweitzer. Exploration of graphs with excluded minors. European Symposium on Algorithms, 2023. Google Scholar
  3. Hans L. Bodlaender, Matthew Johnson, Barnaby Martin, Jelle J. Oostveen, Sukanya Pandey, Daniel Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs IV: the Steiner forest problem. CoRR, abs/2305.01613, 2023. URL: https://doi.org/10.48550/arXiv.2305.01613.
  4. Michael Bussieck. Optimal lines in public rail transport. PhD thesis, Technische Universität Braunschweig, 1998. Google Scholar
  5. Leizhen Cai. NP-completeness of minimum spanner problems. Discret. Appl. Math., 48(2):187-194, 1994. URL: https://doi.org/10.1016/0166-218X(94)90073-6.
  6. Guy Desaulniers and Mark D. Hickman. Public transit. Handbooks in operations research and management science, 14:69-127, 2007. URL: https://doi.org/10.1016/S0927-0507(06)14002-5.
  7. John Ellis and Robert Warren. Lower bounds on the pathwidth of some grid-like graphs. Discrete Applied Mathematics, 156(5):545-555, 2008. URL: https://doi.org/10.1016/j.dam.2007.02.006.
  8. Ehab S. Elmallah and Charles J. Colbourn. Optimum communication spanning trees in series-parallel networks. SIAM J. Comput., 14(4):915-925, 1985. URL: https://doi.org/10.1137/0214064.
  9. Andrés Fielbaum, Sergio Jara-Diaz, and Antonio Gschwender. A parametric description of cities for the normative analysis of transport systems. Networks and Spatial Economics, 17:343-365, 2017. Google Scholar
  10. Markus Friedrich, Maximilian Hartl, Alexander Schiewe, and Anita Schöbel. Integrating passengers' assignment in cost-optimal line planning. In 17th workshop on algorithmic approaches for transportation modelling, optimization, and systems (ATMOS 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  11. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  12. Elisabeth Gassner. The steiner forest problem revisited. J. Discrete Algorithms, 8(2):154-163, 2010. URL: https://doi.org/10.1016/j.jda.2009.05.002.
  13. Philine Gattermann, Jonas Harbering, and Anita Schöbel. Line pool generation. Public Transport, 9:7-32, 2017. Google Scholar
  14. Valérie Guihaire and Jin-Kao Hao. Transit network design and scheduling: A global review. Transportation Research Part A: Policy and Practice, 42(10):1251-1273, 2008. URL: https://doi.org/10.1016/j.tra.2008.03.011.
  15. Anupam Gupta and Amit Kumar. Greedy algorithms for steiner forest. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 871-878. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746590.
  16. Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2023. URL: https://www.gurobi.com.
  17. Refael Hassin and Arie Tamir. Improved complexity bounds for location problems on the real line. Operations Research Letters, 10(7):395-402, 1991. Google Scholar
  18. T. C. Hu. Optimum communication spanning trees. SIAM J. Comput., 3(3):188-195, 1974. URL: https://doi.org/10.1137/0203015.
  19. Rolf Hüttmann. Planungsmodell zur Entwicklung von Nahverkehrsnetzen liniengebundener Verkehrsmittel. PhD thesis, Technische Universität Hannover, 1978. URL: https://orlis.difu.de/handle/difu/482691.
  20. David S. Johnson, Jan Karel Lenstra, and A. H. G. Rinnooy Kan. The complexity of the network design problem. Networks, 8(4):279-285, 1978. URL: https://doi.org/10.1002/net.3230080402.
  21. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  22. David Peleg and Alejandro A. Schäffer. Graph spanners. J. Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/jgt.3190130114.
  23. Santiago Valdés Ravelo and Carlos Eduardo Ferreira. A PTAS for the metric case of the optimum weighted source-destination communication spanning tree problem. Theor. Comput. Sci., 771:9-22, 2019. URL: https://doi.org/10.1016/j.tcs.2018.11.008.
  24. Alexander Schiewe, Sebastian Albert, Philine Schiewe, Anita Schöbel, Felix Spühler, and Moritz Stinzendörfer. Documentation for lintim 2022.08, 2022. Google Scholar
  25. Anita Schöbel. Line planning in public transportation: models and methods. OR spectrum, 34(3):491-510, 2012. URL: https://doi.org/10.1007/s00291-011-0251-6.
  26. Bang Ye Wu, Kun-Mao Chao, and Chuan Yi Tang. Approximation algorithms for some optimum communication spanning tree problems. Discret. Appl. Math., 102(3):245-266, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00212-7.
  27. Bang Ye Wu, Kun-Mao Chao, and Chuan Yi Tang. Light graphs with small routing cost. Networks, 39:130-138, 2002. URL: https://doi.org/10.1002/net.10019.
  28. Carlos Armando Zetina, Iván A. Contreras, Elena Fernández, and Carlos Luna-Mota. Solving the optimum communication spanning tree problem. Eur. J. Oper. Res., 273(1):108-117, 2019. URL: https://doi.org/10.1016/j.ejor.2018.07.055.
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