Dynamic graph generation for the shortest path problem in time expanded networks

In discrete optimization problems the progress of objects over time is frequently modeled by shortest path problems in time expanded networks, but longer time spans or finer time discretizations quickly lead to problem sizes that are intractable in practice. In convex relaxations the arising shortes...

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Bibliographic Details
Published in:Mathematical programming Vol. 143; no. 1-2; pp. 257 - 297
Main Authors: Fischer, Frank, Helmberg, Christoph
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2014
Springer Nature B.V
Subjects:
Algorithms
Analysis
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Corridors
Full Length Paper
Graph theory
Graphs
Lagrange multiplier
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Studies
Theoretical
Large scale optimization
90B10
Time expanded networks
Column generation
90C35
Lagrangian relaxation
Dynamic graph generation
90C06
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:In discrete optimization problems the progress of objects over time is frequently modeled by shortest path problems in time expanded networks, but longer time spans or finer time discretizations quickly lead to problem sizes that are intractable in practice. In convex relaxations the arising shortest paths often lie in a narrow corridor inside these networks. Motivated by this observation, we develop a general dynamic graph generation framework in order to control the networks’ sizes even for infinite time horizons. It can be applied whenever objects need to be routed through a traffic or production network with coupling capacity constraints and with a preference for early paths. Without sacrificing any information compared to the full model, it includes a few additional time steps on top of the latest arcs currently in use. This “frontier” of the graphs can be extended automatically as required by solution processes such as column generation or Lagrangian relaxation. The corresponding algorithm is efficiently implementable and linear in the arcs of the non-time-expanded network with a factor depending on the basic time offsets of these arcs. We give some bounds on the required additional size in important special cases and illustrate the benefits of this technique on real world instances of a large scale train timetabling problem.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0610-3