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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Veröffentlicht in:	Mathematical programming Jg. 143; H. 1-2; S. 257 - 297
Hauptverfasser:	Fischer, Frank, Helmberg, Christoph
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2014 Springer Nature B.V
Schlagworte:	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
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Abstract	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.
AbstractList	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. 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.[PUBLICATION ABSTRACT]
Author	Fischer, Frank Helmberg, Christoph
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Cites_doi	10.1287/opre.1060.0374 10.1016/j.dam.2005.05.026 10.1007/s10288-007-0037-5 10.1007/BF02216922 10.1002/net.10072 10.1016/j.dam.2009.04.007 10.1016/j.trb.2009.07.007 10.1016/S0167-5060(08)70356-X 10.1007/s00291-009-0189-0 10.1016/j.ejor.2011.11.003 10.1002/net.20131 10.1287/opre.50.5.851.362 10.1145/1645953.1646063 10.1007/978-3-540-76796-1_21 10.1007/978-3-540-72845-0_5
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SubjectTerms	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
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Title	Dynamic graph generation for the shortest path problem in time expanded networks
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