Michael Gatto

The computational complexity of delay management

Delay management for public transport consists of deciding whether vehicles should wait for delayed transferring passengers, with the objective of minimizing the overall passenger discomfort. This paper classifies the computational complexity of delay management problems with respect to various structural parameters, such as the maximum number of passenger transfers, the graph topology, and the capability of trains to reduce delays. Our focus is to distinguish between polynomially solvable and nP-complete problem variants. To that end, we show that even fairly restricted versions of the delay management problem are hard to solve.

Railway Delay Management: Exploring Its Algorithmic Complexity

We consider delay management in railway systems. Given delayed trains, we want to find a waiting policy for the connecting trains minimizing the weighted total passenger delay. If there is a single delayed train and passengers transfer at most twice along fixed routes, or if the railway network has a tree structure, the problem can be solved by reduction to min-cut problems. For delayed passenger flows on a railway network with a path structure, the problem can be solved to optimality by dynamic programming. If passengers are allowed to adapt their route to the waiting policy, the decision problem is strongly \(\mathcal{NP}\)-complete.

Shunting for Dummies: An Introductory Algorithmic Survey

In this survey we present a selection of commonly used and new train classification methods from an algorithmic perspective.

Online delay management on a single train line

We provide competitive analyses for the online delay management problem on a single train line. The passengers that want to connect to the train line might arrive delayed at the connecting stations, and these delays happen in an online setting. Our objective is to minimize the total passenger delay on the train line. We relate this problem to the Ski-Rental problem and present a family of 2-competitive online algorithms. Further, we show that no online algorithm for this problem can be better than Golden Ratio competitive, and that no online algorithm can be competitive if the objective accounts only for the optimizable passenger delay.

Optimizing the Cargo Express Service of Swiss Federal Railways

The Cargo Express service of Swiss Federal Railways (SBB Cargo) offers fast overnight transportation of goods between selected train stations in Switzerland and is operated as a hub-and-spoke system with two hubs. We present three different models for planning the operation of this service as a whole. All models capture the underlying optimization problem with a high level of detail: Traffic routing, train routing, makeup, scheduling, and locomotive assignment are all addressed. At the same time we respect hard constraints like tight service time windows and train capacities, and we avoid hub overloading. We describe our approaches for obtaining provably good quality solutions. Our algorithmic techniques involve branch-and-cut, branch-and-price, and problem-specific exact and heuristic acceleration methods. We conclude our study with computational results on realistic data.

On robust online scheduling algorithms

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs’ processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present bounds on the worst case ratio of two makespans. For the first makespan, we let the online algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job’s processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from the competitive ratio of the online algorithm tells us about the “price of perturbations”. We analyze this setting for Graham’s algorithm, and among other bounds show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job. We complement these results by providing lower bounds for any online algorithm in this setting. Finally, we propose a risk-aware online algorithm tailored for the possible bounded increase of the processing time of one job, and we show that this algorithm can be worse than Graham’s algorithm in some cases.

Stability of networks in stretchable graphs

In classic optimization theory, the concept of stability refers to the study of how much and in which way the optimal solutions of a given minimization problem Π can vary as a function of small perturbations of the input data. Motivated by congestion problems arising in shortest-path based communication networks, in this paper we restrict ourselves to the case in which Π is actually a network design problem on a given graph G=(V,E,w) of |V|=n nodes, |E|=m edges, and with a positive real weight w(e) on each edge e∈E. We focus on a subclass of perturbations, that we call stretching perturbations, in which the weights of the edges of G can be increased by at most a fixed multiplicative real factor λ≥1. For this class of perturbations, we address the problem of computing the stability number of any given subgraph H of G containing at least an optimal solution of Π, namely the maximum stretching factor for which H keeps on maintaining an optimal solution. Furthermore, given a stretching factor λ, we study the problem of constructing a minimal subgraph of G with stability number greater or equal to λ. We develop a general technique to solve both problems. By applying this technique to the minimum spanning tree and the single-source shortest paths tree (SPT) problems, we obtain

Railway Delay Management

{\cal O}(m\alpha(m,n))

On the Impact of Uncertainty on some Optimization Problems

and

Optimization of a Railway Hub-and-Spoke System

{\cal O}(mn(m+n \log n))