A Matching Approach for Periodic Timetabling

A Matching Approach for Periodic Timetabling∗ Julius Pätzold1 and Anita Schöbel2 1 Institut für Numerische und Angewandte Mathematik, Georg August University Göttingen, Göttingen, Germany Institut für Numerische und Angewandte Mathematik, Georg August University Göttingen, Göttingen, Germany 2 Abstract The periodic event scheduling problem (PESP) is a well studied problem known as intrinsically hard, but with important applications mainly for finding good timetables in public transportation. In this paper we consider PESP in public transportation, but in a reduced version (r-PESP) in which the driving and waiting times of the vehicles are fixed to their lower bounds. This results in a still NP-hard problem which has less variables, since only one variable determines the schedule for a whole line. We propose a formulation for r-PESP which is based on scheduling the lines. This enables us on the one hand to identify a finite candidate set and an exact solution approach. On the other hand, we use this formulation to derive a matching-based heuristic for solving PESP. Our experiments on close to real-world instances from LinTim show that our heuristic is able to compute competitive timetables in a very short runtime. 1998 ACM Subject Classification G.1.6 Optimization, G.2.2 Graph Theory, G.2.3 Applications Keywords and phrases PESP, Timetabling, Public Transport, Matching, Finite Dominating Set Digital Object Identifier 10.4230/OASIcs.ATMOS.2016.1 1 PESP: The Periodic Event Scheduling Problem The Periodic Event Scheduling Problem (PESP) in which events have to be scheduled periodically is a complex and well-known discrete problem with interesting real-world applications. It has been introduced in [17]. The PESP is known to be NP hard - in fact, even finding a feasible solution is so. The PESP can be formulated as linear mixed-integer program and has been extensively studied. Still, even heuristics are rare and suffer under high empirical run times. Nevertheless using constraint programming techniques, [7] were able to support the decision process of the Netherlands Railway (NS) using the PESP model, and the basic concept of the 2005 timetable of Berlin Underground has been computed in [9]. Solution approaches include constraint generation [14], techniques using the cycle space (see [11, 16, 8]), or the modulo-simplex heuristic [12, 3]. Recently SAT-solvers proved to be successful for solving the PESP [4]. Under research is the construction of timetables under uncertainty, see, e.g., [6, 1]. We start by giving the mathematical formulation of PESP, its interpretation in the context of public transportation will be provided in Section 2. Let an event-activity network N = (E, A) with nodes (or events) E and directed arcs (or activities) A be given. We want to assign a time πi to every event i ∈ E. For setting up feasibility constraints, we furthermore ∗ This work was partially supported by DFG under grant SCHO1140/8-1 © Julius Pätzold and Anita Schöbel; licensed under Creative Commons License CC-BY 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS’16). Editors: Marc Goerigk and Renato Werneck; Article No. 1; pp. 1:1–1:15 Open Access Series in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 1:2 A Matching Approach for Periodic Timetabling assume time spans ∆a = [La , Ua ] with a lower bound La and an upper bound Ua for all activities a ∈ A, and weights wa which represent the importance of activity a ∈ A. Finally, we need a period T ∈ N. An instance I of PESP is hence given by N , w, L, U, T . Defining [x]T := min{x − zT : z ∈ Z, x − zT ≥ 0}, PESP can be formulated as X (PESP) min wa [πj − πi − La ]T a=(i,j)∈A s.t. [πj − πi − La ]T ∈ [0, Ua − La ] for all a ∈ A πi ∈ {0, 1, . . . , T − 1} for all i ∈ E. The variables πi assign a point of time to each event i ∈ E. This time is usually assumed to be integer (in minutes) and takes only values in {0, 1, . . . , T − 1} since it is repeated periodically with a period of T . Note that the PESP only looks at the differences of the π values, hence one of the variables can always be fixed, e.g., π1 := 0. The objective function minimizes the sum of slack times over all activities of the resulting periodic schedule while the constraints ensure that the minimal duration La and maximal duration Ua of all activities a = (i, j) ∈ A are respected by the periodic schedule. Note that [πj − πi − La ]T ∈ [0, Ua − La ] is equivalent to La ≤ πj − πi + za T ≤ Ua for some integer za ∈ Z which can be used to linearize the formulation given above to receive a linear integer program. For details on the periodicity and the meaning of the time spans ∆a we refer to the extensive literature on PESP. Our contribution. In this paper we study the PESP in the context of its main application, namely for timetabling in public transportation. We use the special underlying structure of the event-activity network to design an exact and a heuristic approach for solving the PESP in this case. 2 r-PESP: The reduced periodic event scheduling problem in public transportation We first repeat how the event-activity network is constructed for the case of periodic timetabling in public transportation. Given a set of traffic lines L, the event-activity network N = (E, A) consists of nodes E = Earr ∪ Edep which are called arrival and departure events and of edges A = Adrive ∪ Await ∪ Atrans called driving activities, waiting activities and transfer activities. These are constructed as follows (see, e.g., [11, 8]): Let l ∈ L be a line passing through stations s1 , s2 , . . . , sp . Such a line corresponds to p − 1 arrival and to p − 1 departure events (s1 , l, dep), (s2 , l, arr), (s2 , l, dep), . . . , (sp , l, arr). A departure event (si , l, dep) and its consecutive arrival event (si+1 , l, arr) on the same line l at its next station are linked by a directed driving activity. Waiting activities link an arrival event of a line (si , l, arr) and its consecutive departure event (si , l, dep) at the same station. Transfer activities connect an arrival event (s, l, arr) of one line l at some station s to a departure event (s, k, dep) of another line k at the same station s if a transfer for the passengers should be possible here. J. Pätzold and A. Schöbel 1:3 Note that in railway applications also headway activities are needed which ensure a minimal distance between two consecutive trains on the same piece of infrastructure. In the PESP formulation, the La describe lower bounds on the activities, i.e., the minimal driving time for driving activities, the minimal dwell time at stations for waiting activities and the minimal time needed for a transfer (i.e., getting off the train, changing the platform and boarding the next train) for the passengers for transfer activities. The weights wa give the number of passengers who use activity a ∈ A. Minimizing the sum o (...truncated)