Finding short integral cycle bases for cyclic timetabling

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 forcyclic timetabling. © Springer-Verlag 2003.