The Second Chvatal Closure Can Yield Better Railway Timetables

The Second Chvátal Closure Can Yield Better Railway Timetables⋆ Christian Liebchen and Elmar Swarat Technische Universität Berlin, Institute of Mathematics, Straße des 17. Juni 136, D-10623 Berlin, Germany {liebchen,swarat}@math.tu-berlin.de Abstract. We investigate the polyhedral structure of the Periodic Event Scheduling Problem (PESP), which is commonly used in periodic railway timetable optimization. This is the first investigation of Chvátal closures and of the Chvátal rank of PESP instances. In most detail, we first provide a PESP instance on only two events, whose Chvátal rank is very large. Second, we identify an instance for which we prove that it is feasible over the first Chvátal closure, and also feasible for another prominent class of known valid inequalities, which we reveal to live in much larger Chvátal closures. In contrast, this instance turns out to be infeasible already over the second Chvátal closure. We obtain the latter result by introducing new valid inequalities for the PESP, the multi-circuit cuts. In the past, for other classes of valid inequalities for the PESP, it had been observed that these do not have any effect in practical computations. In contrast, the new multi-circuit cuts that we are introducing here indeed show some effect in the computations that we perform on several real-world instances – a positive effect, in most of the cases. 1 Introduction It has been only recently that combinatorial optimization entered the practice of service design in public transport. The 2005 timetable of Berlin Underground is the first optimized timetable that was put into service [9]. It had been computed with integer programming techniques, namely profiting from several different classes of valid inequalities. Today, also the Dutch railways are operating a timetable that was designed with the help of techniques from combinatorial optimization and constraint programming [7]. Both projects build upon the Periodic Event Scheduling Problem (PESP). The PESP, in its pure formulation of a feasibility problem, had been introduced by Serafini and Ukovich [18] and it generalizes the vertex coloring problem. In particular, for the two most natural optimization problems that are investigated on top of the PESP, MAXSNP-hardness has been established [8, 9]. In practice, this results in the following typical behavior of MIP solvers on medium to large sized instances. Known valid inequalities are able to close 60–90% of the initial gap between the integer optimum value and the optimum value of the LP relaxation. Still, solving this tightened IP risks to take several hours, if it is solvable at all. ⋆ Supported by the DFG Research Center M ATHEON in Berlin. ATMOS 2008 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems http://drops.dagstuhl.de/opus/volltexte/2008/1580 2 Christian Liebchen and Elmar Swarat There are of course much larger transportation networks in practice, which are beyond the computational limits of the methods that were used so far. As a consequence, at present there are several other research groups trying to tackle the periodic railway timetabling problem, and they are sharing the PESP as their model of choice [2, 17, 19]. For instance, Villumsen put the polyhedral approach that was suggested by Lindner [14] into practical computations for the commuter train network of Copenhagen. Unfortunately, he had to make the observation that “the chain cuts [14] have no effect on the solution” [19]. This is one motivation for us to have a closer look at the polyhedral structure of the feasible region of PESP instances. We do so by following the methodology that has been suggested recently by Fischetti and Lodi [6] for optimizing over the first Chvátal closure. Notice that one of the first instances to which they applied their method was the “hard MIPLIB instance timtab1”, which is in fact a PESP model [10]. As a motivation, we first generalize an infeasible PESP instance – which is due to Lindner [14] – to a family of instances that are defined on wheel graphs. In Section 6 we will prove that these instances are feasible over the first Chvátal closure. Still worse, even the change-cycle inequalities that have been introduced by Nachtigall [15], of which in Section 4 we prove that, in general, they lie in much larger Chvátal closures, are not suited to certify infeasibility. Nevertheless, the techniques of Fischetti and Lodi suggested that these particular instances might be infeasible already over the second Chvátal closure. Indeed, by exploiting problem-specific insight, in the second Chvátal closure we identify general new valid inequalities for the PESP (Section 5) by which we prove that these particular instances are infeasible. We call these new inequalities the multi-circuit cuts. In Section 7 we add multi-circuit cuts to the IP formulations of several timetabling instances that we took from practice. Although we have to admit that the results are not fully striking, on many instances we observe a perceptible speed-up in the solution time. In turn, on more complex instances, for which up to now no optimal solution has been found, our new cuts from the second Chvátal closure might indeed yield better railway timetables. 2 An IP for PESP Initially, the Periodic Event Scheduling Problem (PESP, [18]) has been stated as a pure feasibility problem. We are given a directed graph D = (V, A), which may feature (anti-) parallel arcs. For each arc a, there are defined some lower bound ℓa and some upper bound ua . The PESP then asks whether for the given fixed period time T , the instance admits a (periodically) feasible node potential π ∈ [0, T )V , i.e., (πj − πi − ℓa ) mod T ≤ ua − ℓa , ∀a = (i, j) ∈ A. (1) In a railway timetabling context, the value T is the period time of the railway system, e.g., 60 minutes. A node i represents an arrival or departure of some specific directed line in the network, and we must assign a time value πi to this event. For instance, The Second Chvátal Closure Can Yield Better Railway Timetables 3 in the current timetable, the direct ICE trains from Berlin to Karlsruhe leave Berlin main station 33 minutes past the hour. Finally, in the constraint parameters ℓ and u one may encode lower and upper bounds on time durations to ensure safety requirements, transfer quality requirements, as well as many other features [11]. In a mixed-integer linear programming formulation, the modulo-operator in (1) is resolved by introducing integer variables pa for the arcs, which we denote periodical offsets. Furthermore, we penalize any slack on the lower bounds ℓa in a linear objective function, P min a=(i,j)∈A wa (πj − πi + T pa ) s.t. πj − πi + T pa ≥ ℓa , ∀a = (i, j) ∈ A (2) πj − πi + T pa ≤ ua , ∀a = (i, j) ∈ A πi ∈ [0, T ), ∀i ∈ V pa ∈ Z, ∀a ∈ A. Other formulations for this problem had been stated in terms of so-called tension variables ya = πj − πi , or even periodi (...truncated)