Algorithms and uncertainty sets for data-driven robust shortest path problems

Abstract

We consider robust shortest path problems, where the aim is to find a path that optimizes the worst-case performance over an uncertainty set containing all relevant scenarios for arc costs. The usual approach for such problems is to assume this uncertainty set given by an expert who can advise on the shape and size of the set.Following the idea of data-driven robust optimization, we instead construct a range of uncertainty sets from the current literature based on real-world traffic measurements provided by the City of Chicago. We then compare the performance of the resulting robust paths within and outside the sample, which allows us to draw conclusions what the most suited uncertainty set is.Based on our experiments, we then focus on ellipsoidal uncertainty sets, and develop a new solution algorithm that significantly outperforms a stateof-the-art solver.robust shortest path problems, on the other hand, are NP-hard (see [20]), and real-time information has not been an option.To formulate a robust problem, it is necessary to have a description of all possible and relevant scenarios that the solution should prepare against, the so-called uncertainty set. We refer to the surveys [1,16,17] for a general overview on the topic. The current literature on robust shortest paths usually assumes this set to be given, by some mixture of data-preprocessing and expert knowledge that is not part of the study. This means that different types of sets have been studied (compare, e.g., [18,9]), but it has been impossible to address the question which would be the "right" choice.A recent paradigm shift is data-driven robust optimization (see [6]), where building the uncertainty set from raw observations is part of the robust optimization problem. This paper is the first to follow such a perspective for shortest path problems. Based on real-world observations from the City of Chicago, we build a range of uncertainty sets, calculate the corresponding robust solutions, and perform an in-depth analysis of their performance. This allows us to give an indication which set is actually suitable for our application, and which are not.In the second part of this paper, we then focus on the case of ellipsoidal uncertainty, and provide a branch-and-bound algorithm that is able to solve instances considerably faster than an off-the-shelf solver.Parts of this paper were previously published as a conference paper in [15]. In comparison, we provide a completely new set of experimental results based on an observation period of 46 days (instead of one single day), which leads to a more detailed insight into the performance of different uncertainty sets. Furthermore, we provide a new analysis for axis-parallel ellipsoidal uncertainty sets, including an efficient branch-and-bound algorithm that is able to outperform Cplex by several orders of magnitude, pushing robust shortest paths towards applicability in real-time navigation systems.The remainder of the paper is structured as follows. In Section 2 we briefly introduce the robust shortest path probl...

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“…Parameter λ represents the size of the uncertainty set, in relation to the observed scenarios. For a formal parameter definition, we refer to [CDG19].…”

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