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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

Arc flow formulations based on dynamic programming: Theoretical foundations and applications

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Author(s):
de Lima, Vinicius L. [1] ; Alves, Claudio [2] ; Clautiaux, Francois [3] ; Iori, Manuel [4] ; Valerio de Carvalho, Jose M. [2]
Total Authors: 5
Affiliation:
[1] Univ Estadual Campinas, Inst Comp, Campinas - Brazil
[2] Univ Minho, Ctr Algoritmi, Escola Engn, Braga - Portugal
[3] Univ Bordeaux, IMB UMR CNRS 5251, Inria Bordeaux Sud Ouest, Bordeaux - France
[4] Univ Modena & Reggio Emilia, DISMI, Modena - Italy
Total Affiliations: 4
Document type: Review article
Source: European Journal of Operational Research; v. 296, n. 1, p. 3-21, JAN 1 2022.
Web of Science Citations: 0
Abstract

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For N P-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation. In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing. (c) 2021 Elsevier B.V. All rights reserved. (AU)

FAPESP's process: 17/11831-1 - Algorithms and models for cutting and packing problems
Grantee:Vinícius Loti de Lima
Support type: Scholarships in Brazil - Doctorate (Direct)
FAPESP's process: 19/12728-5 - The study of theoretical and practical combinatorial optimization problems applied on real scenarios
Grantee:Flávio Keidi Miyazawa
Support type: Research Grants - Visiting Researcher Grant - International