Abstract
Cyclic timetabling for public transportation companies is usually modeled by the periodic event scheduling problem. To obtain a mixed-integer programming formulation, artificial integer variables have to be introduced. There are many ways to define these integer variables.
We show that the minimal number of integer variables required to encode an instance is achieved by introducing an integer variable for each element of some integral cycle basis of the directed graph D=(V,A) defining the periodic event scheduling problem. Here, integral means that every oriented cycle can be expressed as an integer linear combination.
The solution times for the originating application vary extremely with different integral cycle bases. Our computational studies show that the width of integral cycle bases is a good empirical measure for the solution time of the MIP. Integral cycle bases permit a much wider choice than the standard approach, in which integer variables are associated with the co-tree arcs of some spanning tree. To formulate better solvable integer programs, we present algorithms that construct good integral cycle bases. To that end, we investigate subsets and supersets of the set of integral cycle bases. This gives rise to both, a compact classification of directed cycle bases and notable reductions of running times for cyclic timetabling.
Supported by the DFG Research Center “Mathematics for key technologies” in Berlin
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References
Amaldi, E.: Personal Communication. Politecnico di Milano, Italy (2003)
Berge, C.: The Theory of Graphs and its Applications. John Wiley & Sons, Chichester (1962)
Berger, F.: Minimale Kreisbasen in Graphen. Lecture on the annual meeting of the DMV in Halle, Germany (2002)
Champetier, C.: On the Null-Homotopy of Graphs. Discrete Mathematics 64, 97–98 (1987)
CPLEX 8.0 ILOG SA, France (2002), http://www.ilog.com/products/cplex
Deo, N., Kumar, N., Parsons, J.: Minimum-Length Fundamental-Cycle Set Problem: A New Heuristic and an SIMD Implementation. Technical Report CS-TR-95-04, University of Central Florida, Orlando (1995)
Deo, N., Prabhu, M., Krishnamoorthy, M.S.: Algorithms for Generating Fundamental Cycles in a Graph. ACM Transactions on Mathematical Software 8, 26–42 (1982)
Gleiss, P.: Short Cycles. Ph.D. Thesis, University of Vienna, Austria (2001)
Golynski, A., Horton, J.D.: A Polynomial Time Algorithm to Find the Minimum Cycle Basis of a Regular Matroid. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, p. 200. Springer, Heidelberg (2002)
Hartvigsen, D., Zemel, E.: Is Every Cycle Basis Fundamental? Journal of Graph Theory 13, 117–137 (1989)
Horton, J.D.: A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing 16, 358–366 (1987)
Krista, M.: Verfahren zur Fahrplanoptimierung dargestellt am Beispiel der Synchronzeiten (Methods for Timetable Optimization Illustrated by Synchronous Times). Ph.D. Thesis, Technical University Braunschweig, Germany. (1996) in German
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)
Leydold, J., Stadler, P.F.: Minimal Cycle Bases of Outerplanar Graphs. The Electronic Journal of Combinatorics 5, #16 (1998)
Liebchen, C., Peeters, L.: On Cyclic Timetabling and Cycles in Graphs. Technical Report 761/2002, TU Berlin (2002)
Liebchen, C., Peeters, L.: Some Practical Aspects of Periodic Timetabling. In: Chamoni, P., et al. (eds.) Operations Research 2001. Springer, Heidelberg (2002)
Nachtigall, K.: A Branch and Cut Approach for Periodic Network Programming. Hildesheimer Informatik-Berichte, 29 (1994)
Nachtigall, K.: Cutting planes for a polyhedron associated with a periodic network. DLR Interner Bericht, 17 (1996)
Nachtigall, K.: Periodic network optimization with different arc frequencies. Discrete Applied Mathematics 69, 1–17 (1996)
Odijk, M.: Railway Timetable Generation. Ph.D. Thesis, TU Delft, The Netherlands (1997)
de Pina, J.C.: Applications of Shortest Path Methods. Ph.D. Thesis, University of Amsterdam, The Netherlands (1995)
Schrijver, A.: Theory of Linear and Integer Programming, 2nd edn. Wiley, Chichester (1998)
Serafini, P., Ukovich, W.: A mathematical model for periodic scheduling problems. SIAM Journal on Discrete Mathematics 2, 550–581 (1989)
Whitney, H.: On the Abstract Properties of Linear Dependence. American Journal of Mathematics 57, 509–533 (1935)
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Liebchen, C. (2003). Finding Short Integral Cycle Bases for Cyclic Timetabling. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_64
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DOI: https://doi.org/10.1007/978-3-540-39658-1_64
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