Cost Projection Methods for the Shortest Path Problem with Crossing Costs

Blanco, Marco; Borndörfer, Ralf; Hoang, Nam-Dung; Kaier, Anton; Casas, Pedro Maristany de las; Schlechte, Thomas; Schlobach, Swen

doi:10.4230/oasics.atmos.2017.15

Cited by 5 publications

(1 citation statement)

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“…Delling, Pajor, and Werneck (2015) used their well-known RAPTOR (RAP) algorithm to compute journeys that touch the smallest number of fare zones. A variant of zone-based fare systems in the context of overflight costs was studied by Blanco et al (2016) and Blanco et al (2017). Recently, Gündling (2020) considered price-optimized routing in intermodal transportation.…”

Section: Related Literaturementioning

confidence: 99%

Price Optimal Routing in Public Transportation

Euler¹,

Borndörfer²

2022

Preprint

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With the development of fast routing algorithms for public transit the optimization of criteria other than arrival time has moved into the spotlight. Due to the often intricate nature of fare regulations, the price of a journey, while certainly being one of the most interesting criteria, is seldom considered. In this work, we present a novel framework to mathematically describe the fare systems of local public transit companies. The model allows us to solve the price-sensitive earliest arrival problem (PSEAP) even if prices depend on a number of parameters and non-linear conditions. We base our approach on conditional fare networks (CFN). A CFN consists of a ticket graph modeling tickets and the relations between them and a ticket transition function defined over a partially ordered monoid and a set of fare events. Typically, shortest path algorithms rely on the subpath optimality property which is usually lost when dealing with intricate fare systems. We show how subpath optimality can be restored by relaxing domination rules for tickets depending on the structure of the ticket graph. An exemplary model for the fare system of Mitteldeutsche Verkehrsbetriebe (MDV) is provided. PSEAP is NP-hard, even when considering a single-criterion version that optimizes for price only. Instances on finite monoids are, however, polynomial time solvable when the monoid is considered part of the encoding length. By integrating our framework in the multi-criteria RAPTOR algorithm, we provide an algorithm for PSEAP and assess its performance on data obtained from MDV. We introduce two simple speed-up techniques that in combination with the recently introduced Tight-BMRAPTOR pruning scheme allow us to solve PSEAP in mere milliseconds on our data set.

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Section: Related Literaturementioning

confidence: 99%

Price Optimal Routing in Public Transportation

Euler¹,

Borndörfer²

2022

Preprint

Self Cite

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Error Bounds for Discrete-Continuous Free Flight Trajectory Optimization

Borndörfer

Danecker

Weiser

2023

J Optim Theory Appl

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Two-stage methods addressing continuous shortest path problems start local minimization from discrete shortest paths in a spatial graph. The convergence of such hybrid methods to global minimizers hinges on the discretization error induced by restricting the discrete global optimization to the graph, with corresponding implications on choosing an appropriate graph density. A prime example is flight planning, i.e., the computation of optimal routes in view of flight time and fuel consumption under given weather conditions. Highly efficient discrete shortest path algorithms exist and can be used directly for computing starting points for locally convergent optimal control methods. We derive a priori and localized error bounds for the flight time of discrete paths relative to the optimal continuous trajectory, in terms of the graph density and the given wind field. These bounds allow designing graphs with an optimal local connectivity structure. The properties of the bounds are illustrated on a set of benchmark problems. It turns out that localization improves the error bound by four orders of magnitude, but still leaves ample opportunities for tighter error bounds by a posteriori estimators.

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