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DOI: 10.4230/OASIcs.ATMOS.2018.5

Fast Robust Shortest Path Computations

Authors: Christoph Hansknecht, Alexander Richter, and Sebastian Stiller

Published in: OASIcs, Volume 65, 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)

Abstract

We develop a fast method to compute an optimal robust shortest path in large networks like road networks, a fundamental problem in traffic and logistics under uncertainty. In the robust shortest path problem we are given an s-t-graph D(V,A) and for each arc a nominal length c(a) and a maximal increase d(a) of its length. We consider all scenarios in which for the increased lengths c(a) + bar{d}(a) we have bar{d}(a) <= d(a) and sum_{a in A} (bar{d}(a)/d(a)) <= Gamma. Each path is measured by the length in its worst-case scenario. A classic result [Bertsimas and Sim, 2003] minimizes this path length by solving (|A| + 1)-many shortest path problems. Easily, (|A| + 1) can be replaced by |Theta|, where Theta is the set of all different values d(a) and 0. Still, the approach remains impractical for large graphs. Using the monotonicity of a part of the objective we devise a Divide and Conquer method to evaluate significantly fewer values of Theta. This methods generalizes to binary linear robust problems. Specifically for shortest paths we derive a lower bound to speed-up the Divide and Conquer of Theta. The bound is based on carefully using previous shortest path computations. We combine the approach with non-preprocessing based acceleration techniques for Dijkstra adapted to the robust case. In a computational study we document the value of different accelerations tried in the algorithm engineering process. We also give an approximation scheme for the robust shortest path problem which computes a (1 + epsilon)-approximate solution requiring O(log(d^ / (1 + epsilon))) computations of the nominal problem where d^ := max d(A) / min (d(A)\{0}).

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Christoph Hansknecht, Alexander Richter, and Sebastian Stiller. Fast Robust Shortest Path Computations. In 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018). Open Access Series in Informatics (OASIcs), Volume 65, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hansknecht_et_al:OASIcs.ATMOS.2018.5,
  author =	{Hansknecht, Christoph and Richter, Alexander and Stiller, Sebastian},
  title =	{{Fast Robust Shortest Path Computations}},
  booktitle =	{18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)},
  pages =	{5:1--5:21},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-096-5},
  ISSN =	{2190-6807},
  year =	{2018},
  volume =	{65},
  editor =	{Bornd\"{o}rfer, Ralf and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2018.5},
  URN =		{urn:nbn:de:0030-drops-97100},
  doi =		{10.4230/OASIcs.ATMOS.2018.5},
  annote =	{Keywords: Graph Algorithms, Shortest Paths, Robust Optimization}
}

Document

DOI: 10.4230/OASIcs.ATMOS.2012.58

Multi-Dimensional Commodity Covering for Tariff Selection in Transportation

Authors: Felix G. König, Jannik Matuschke, and Alexander Richter

Published in: OASIcs, Volume 25, 12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2012)

Abstract

In this paper, we study a multi-dimensional commodity covering problem, which we encountered as a subproblem in optimizing large scale transportation networks in logistics. The problem asks for a selection of containers for transporting a given set of commodities, each commodity having different extensions of properties such as weight or volume. Each container can be selected multiple times and is specified by a fixed charge and capacities in the relevant properties. The task is to find a cost minimal collection of containers and a feasible assignment of the demand to all selected containers. From theoretical point of view, by exploring similarities to the well known SetCover problem, we derive NP-hardness and see that the non-approximability result known for set cover also carries over to our problem. For practical applications we need very fast heuristics to be integrated into a meta-heuristic framework that - depending on the context - either provide feasible near optimal solutions or only estimate the cost value of an optimal solution. We develop and analyze a flexible family of greedy algorithms that meet these challenges. In order to find best-performing configurations for different requirements of the meta-heuristic framework, we provide an extensive computational study on random and real world instance sets obtained from our project partner 4flow AG. We outline a trade-off between running times and solution quality and conclude that the proposed methods achieve the accuracy and efficiency necessary for serving as a key ingredient in more complex meta-heuristics enabling the optimization of large-scale networks.

Cite as

Felix G. König, Jannik Matuschke, and Alexander Richter. Multi-Dimensional Commodity Covering for Tariff Selection in Transportation. In 12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 25, pp. 58-70, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{konig_et_al:OASIcs.ATMOS.2012.58,
  author =	{K\"{o}nig, Felix G. and Matuschke, Jannik and Richter, Alexander},
  title =	{{Multi-Dimensional Commodity Covering for Tariff Selection in Transportation}},
  booktitle =	{12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{58--70},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-45-3},
  ISSN =	{2190-6807},
  year =	{2012},
  volume =	{25},
  editor =	{Delling, Daniel and Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2012.58},
  URN =		{urn:nbn:de:0030-drops-37034},
  doi =		{10.4230/OASIcs.ATMOS.2012.58},
  annote =	{Keywords: Covering, Heuristics, Transportation, Tariff Selection}
}

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Fast Robust Shortest Path Computations

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Multi-Dimensional Commodity Covering for Tariff Selection in Transportation

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