Skip to main content
SpringerLink
Log in

Strong Formulations for the Multi-module PESP and a Quadratic Algorithm for Graphical Diophantine Equation Systems

  • Conference paper
Algorithms – ESA 2010 (ESA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6346))

Included in the following conference series:

Abstract

The Periodic Event Scheduling Problem (PESP) is the method of choice for real-world periodic timetabling in public transport. Its MIP formulation has been studied intensely for the case of uniform modules, i.e., when all events have the same period. In practice, multiple periods are equally important. Yet, the powerful methods developed for uniform modules generally fail for the multi-module case. We analyze a certain type of Diophantine equation systems closely related to the multi-module PESP. Thereby, we identify a structure, so-called sharp trees, that allows to solve the system in \(\mathcal{O}(n^2)\) time if the modules form a linear lattice. Based on this we develop the machinery to solve multi-module PESPs on real-world scale. In our computational results the new MIP-formulations considerably improve the solvability of multi-module PESPs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Conforti, M., Di Summa, M., Wolsey, L.: The mixing set with divisible capacities. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 435–449. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Conforti, M., Zambelli, G.: The mixing set with divisible capacities: A simple approach. Operations Research Letters 37, 379–383 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fischetti, M., Lodi, A.: Optimizing over the first Chvatal closure. Mathematical Programming 110(1), 3–20 (2006)

    Article  MathSciNet  Google Scholar 

  4. Galli, L., Stiller, S.: Strong Formulations for the Multi-module PESP and a Quadratic Algorithm for Graphical Diophantine Equation Systems. COGA Technical Report 009–2010 (2010)

    Google Scholar 

  5. Hassin, R.: A flow algorithm for network synchronization. Operations Research 44, 570–579 (1996)

    Article  MATH  Google Scholar 

  6. Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 767–775. ACM Press, New York (2002)

    Google Scholar 

  7. Köhler, E., Möhring, R., Nökel, K., Wünsch, G.: Optimization of Signalized Traffic Networks. In: Mathematics – Key Technology for the Future, pp. 179–180. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Liebchen, C.: Periodic Timetable Optimization in Public Transport. Ph.D. thesis, Technische Universität Berlin (2006)

    Google Scholar 

  9. Liebchen, C., Swarat, E.: The Second Chvatal Closure Can Yield Better Railway Timetables. In: Proceedings of 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems. Online Schloss Dagstuhl (2008)

    Google Scholar 

  10. Liebchen, C., Proksch, M., Wagner, F.H.: Performance of Algorithms for Periodic Timetable Optimization. In: Computer-aided Systems in Public Transport, pp. 151–180. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  11. Nachtigall, K.: Cutting planes for a polyhedron associated with a periodic network. DLR Technical Report 112-96/17

    Google Scholar 

  12. Odijk, M.: Construction of periodic timetables, Part1: a cutting plane algorithm. TU Delft Technical Report 94-61 (1994)

    Google Scholar 

  13. Odijk, M.: A constraint generation algorithm for the construction of periodic railway timetables. Transportation Research B 30(6), 455–464 (1996)

    Article  Google Scholar 

  14. Peeters, L.: Cyclic Railway Timetable Optimization. Ph.D. thesis, Erasmus University of Rotterdam (2003)

    Google Scholar 

  15. Schrijver, A.: Theory of Linear and Integer Programming. Wiley & Sons, Chichester (1986)

    MATH  Google Scholar 

  16. Serafini, P., Ukovich, W.: A mathematical model for periodic scheduling problems. SIAM Journal on Discrete Mathematics 2(4), 550–581 (1989)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DEIS, Alma Mater University of Bologna, Italy

    Laura Galli

  2. Institut für Mathematik, Technische Universität Berlin, Germany

    Sebastian Stiller

Authors
  1. Laura Galli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Stiller
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands

    Mark de Berg

  2. Institute for Computer Science, J.W. Goethe University, 60325, Frankfurt/Main, Germany

    Ulrich Meyer

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Galli, L., Stiller, S. (2010). Strong Formulations for the Multi-module PESP and a Quadratic Algorithm for Graphical Diophantine Equation Systems. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15775-2_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15774-5

  • Online ISBN: 978-3-642-15775-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation