New Perspectives on PESP: T-Partitions and Separators

Authors Niels Lindner, Christian Liebchen

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Niels Lindner
  • Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany
Christian Liebchen
  • Technical University of Applied Sciences Wildau, Hochschulring 1, 15745 Wildau, Germany


The authors want to thank Ralf Borndörfer for fruitful conversations.

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Niels Lindner and Christian Liebchen. New Perspectives on PESP: T-Partitions and Separators. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019). Open Access Series in Informatics (OASIcs), Volume 75, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In the planning process of public transportation companies, designing the timetable is among the core planning steps. In particular in the case of periodic (or cyclic) services, the Periodic Event Scheduling Problem (PESP) is well-established to compute high-quality periodic timetables. We are considering algorithms for computing good solutions and dual bounds for the very basic PESP with no additional extra features as add-ons. The first of these algorithms generalizes several primal heuristics that have been proposed, such as single-node cuts and the modulo network simplex algorithm. We consider partitions of the graph, and identify so-called delay cuts as a structure that allows to generalize several previous heuristics. In particular, when no more improving delay cut can be found, we already know that the other heuristics could not improve either. This heuristic already had been proven to be useful in computational experiments [Ralf Borndörfer et al., 2019], and we locate it in the more general concept of what we denote T-partitions. With the second of these algorithms we propose to turn a strategy, that has been discussed in the past, upside-down: Instead of gluing together the network line-by-line in a bottom-up way, we develop a divide-and-conquer-like top-down approach to separate the initial problem into two easier subproblems such that the information loss along their cutset edges is as small as possible. We are aware that there may be PESP instances that do not fit well the separator setting. Yet, on the RxLy-instances of PESPlib in our experimental computations, we come up with good primal solutions and dual bounds. In particular, on the largest instance (R4L4), this new separator approach, which applies a state-of-the-art solver as subroutine, is able to come up with better dual bounds than purely applying this state-of-the-art solver in the very same time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Applied computing → Transportation
  • Mathematics of computing → Discrete optimization
  • Mathematics of computing → Integer programming
  • Periodic Event Scheduling Problem
  • Periodic Timetabling
  • Graph Partitioning
  • Graph Separators
  • Balanced Cuts


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  1. Ralf Borndörfer, Niels Lindner, and Sarah Roth. A Concurrent Approach to the Periodic Event Scheduling Problem. Technical Report 19-07, Zuse Institute Berlin, 2019. To appear in RailNorrköping2019 - 8th International Conference on Railway Operations Modelling and Analysis. Google Scholar
  2. Daniel Delling, Daniel Fleischman, Andrew V. Goldberg, Ilya Razenshteyn, and Renato F. Werneck. An exact combinatorial algorithm for minimum graph bisection. Mathematical Programming, 153(2):417-458, November 2015. Google Scholar
  3. M.R. Garey, D.S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. Google Scholar
  4. Marc Goerigk and Christian Liebchen. An Improved Algorithm for the Periodic Timetabling Problem. In ATMOS, volume 59 of OASICS, pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  5. Marc Goerigk and Anita Schöbel. Improving the modulo simplex algorithm for large-scale periodic timetabling. Computers &Operations Research, 40(5):1363-1370, 2013. Google Scholar
  6. Peter Großmann. Satisfiability and Optimization in Periodic Traffic Flow Problems. PhD thesis, TU Dresden, 2016. Google Scholar
  7. Peter Großmann, Steffen Hölldobler, Norbert Manthey, Karl Nachtigall, Jens Opitz, and Peter Steinke. Solving Periodic Event Scheduling Problems with SAT. In He Jiang, Wei Ding, Moonis Ali, and Xindong Wu, editors, Advanced Research in Applied Artificial Intelligence, pages 166-175, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  8. Michael Hamann and Ben Strasser. Graph Bisection with Pareto Optimization. J. Exp. Algorithmics, 23:1.2:1-1.2:34, February 2018. Google Scholar
  9. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer US, Boston, MA, 1972. Google Scholar
  10. George Karypis and Vipin Kumar. A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. Sci. Comput., 20(1):359-392, December 1998. Google Scholar
  11. Leo Kroon, Dennis Huisman, Erwin Abbink, Pieter-Jan Fioole, Matteo Fischetti, Gábor Maróti, Alexander Schrijver, Adri Steenbeek, and Roelof Ybema. The new Dutch timetable: The OR revolution. Interfaces, 39(1):6-17, 2009. Google Scholar
  12. Christian Liebchen. Optimierungsverfahren zur Erstellung von Taktfahrplänen. Master’s thesis, Technical University Berlin, Germany, 1998. In German. Google Scholar
  13. Christian Liebchen. A Cut-Based Heuristic to Produce Almost Feasible Periodic Railway Timetables. In Sotiris E. Nikoletseas, editor, Experimental and Efficient Algorithms, 4th International Workshop, WEA 2005, Santorini Island, Greece, May 10-13, 2005, Proceedings, volume 3503 of Lecture Notes in Computer Science, pages 354-366. Springer, 2005. URL:
  14. Christian Liebchen. Optimization of passenger timetables: Are fully-integrated, regular-interval timetables really always the best? European Rail Technical Review (RTR), 48(4):13-19, 2008. Google Scholar
  15. Christian Liebchen. The first optimized railway timetable in practice. Transportation Science, 42(4):420-435, 2008. Google Scholar
  16. Christian Liebchen and Rolf H Möhring. The Modeling Power of the Periodic Event Scheduling Problem: Railway Timetables—and Beyond. In Algorithmic methods for railway optimization, pages 3-40. Springer, 2007. Google Scholar
  17. Christian Liebchen and Leon Peeters. Integral cycle bases for cyclic timetabling. Discrete Optimization, 6(1):98-109, 2009. Google Scholar
  18. Christian Liebchen and Elmar Swarat. The Second Chvatal Closure Can Yield Better Railway Timetables. In Matteo Fischetti and Peter Widmayer, editors, 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08), volume 9 of OpenAccess Series in Informatics (OASIcs), Dagstuhl, Germany, 2008. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  19. Karl Nachtigall. Periodic Network Optimization and Fixed Interval Timetables. Habilitation thesis, Universität Hildesheim, 2008. Google Scholar
  20. Karl Nachtigall and Jens Opitz. Solving Periodic Timetable Optimisation Problems by Modulo Simplex Calculations. In Matteo Fischetti and Peter Widmayer, editors, 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'08), volume 9 of OpenAccess Series in Informatics (OASIcs), Dagstuhl, Germany, 2008. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  21. Karl Nachtigall and Stefan Voget. A genetic algorithm approach to periodic railway synchronization. Computers & OR, 23(5):453-463, 1996. URL:
  22. Michiel A Odijk. A constraint generation algorithm for the construction of periodic railway timetables. Transportation Research Part B: Methodological, 30(6):455-464, 1996. Google Scholar
  23. Julius Pätzold and Anita Schöbel. A Matching Approach for Periodic Timetabling. In ATMOS' 16, volume 54 of OASICS, pages 1:1-1:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  24. Peter Sanders and Christian Schulz. Think Locally, Act Globally: Highly Balanced Graph Partitioning. In Proceedings of the 12th International Symposium on Experimental Algorithms (SEA'13), volume 7933 of LNCS, pages 164-175. Springer, 2013. Google Scholar
  25. Aaron Schild and Christian Sommer. On Balanced Separators in Road Networks. In Evripidis Bampis, editor, Experimental Algorithms - 14th International Symposium, SEA 2015, Paris, France, June 29 - July 1, 2015, Proceedings, volume 9125 of Lecture Notes in Computer Science, pages 286-297. Springer, 2015. URL:
  26. Alexander Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  27. Paolo Serafini and Walter Ukovich. A mathematical model for periodic scheduling problems. SIAM Journal on Discrete Mathematics, 2(4):550-581, 1989. Google Scholar
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