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Main Title: Continuous-time Dynamic Shortest Paths with Negative Transit Times
Author(s): Koch, Ronald
Nasrabadi, Ebrahim
Type: Research Paper
Abstract: We consider the dynamic shortest path problem in the continuous-time model because of its importance. This problem has been extensively studied in the literature. But so far, all contributions to this problem are based on the assumption that all transit times are strictly positive. However, in order to study dynamic network flows it is essential to support negative transit times since they occur quite naturally in residual networks. In this paper we extend the work of Philpott [SIAM Control Opt., 1994, pp. 538-552] to the case of arbitrary (also negative) transit times. In particular, we study a corresponding linear program in space of measures and characterize its extreme points. We show a one-to-one correspondence between extreme points and dynamic paths. Further, under certain assumptions, we prove the existence of an optimal extreme point to the linear program and establish a strong duality result. We also present counterexamples to show that strong duality only holds under these assumptions.
Subject(s): shortest path problem
linear programming in measure spaces
extreme points
duality theory
measure theory
Issue Date: 2010
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 05C38 Paths and cycles
49J27 Existence theories for problems in abstract spaces
90C46 Optimality conditions and duality in mathematical programming
46E27 Spaces of measures
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2010, 06
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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