IDEAS home Printed from https://ideas.repec.org/a/spr/aqjoor/v23y2025i1d10.1007_s10288-025-00585-z.html
   My bibliography  Save this article

A tutorial on Branch-Price-and-Cut algorithms

Author

Listed:
  • Matteo Petris

    (Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL
    CERMICS, ENPC, Institut Polytechnique de Paris)

  • Claudia Archetti

    (University of Brescia)

  • Diego Cattaruzza

    (University of Udine
    Univ. Lille, CNRS, Inria, Centrale Lille, UMR 9189 CRIStAL)

  • Maxime Ogier

    (Univ. Lille, CNRS, Inria, Centrale Lille, UMR 9189 CRIStAL)

  • Frédéric Semet

    (Univ. Lille, CNRS, Inria, Centrale Lille, UMR 9189 CRIStAL)

Abstract

This paper provides a tutorial on Branch-Price-and-Cut (BPC) algorithms for a generic class of problems whose objective is to find a set of feasible paths in a graph while optimising a given objective function. The tutorial is split into two main parts. First, we describe the building blocks of a BPC algorithm: the Branch-and-Bound algorithm, the column generation procedure and the Branch-and-Cut algorithm. Then, we focus on the description of a BPC algorithm for the class of problems we consider. Precisely, we present the classical and advanced techniques that should be embedded in an efficient algorithm. Particular attention is devoted to the solution of the pricing problem in the case where it is formulated as an Elementary Shortest Path Problem with Resource Constraints. The aim of the tutorial is pedagogical. Hence, its intended reader is someone facing the first implementation of a BPC algorithm. Implementation tips and examples accompany the techniques and concepts to ease their comprehension. Precisely, the examples are based on the Capacitated Vehicle Routing Problem, which is a well-known problem belonging to the class we consider.

Suggested Citation

  • Matteo Petris & Claudia Archetti & Diego Cattaruzza & Maxime Ogier & Frédéric Semet, 2025. "A tutorial on Branch-Price-and-Cut algorithms," 4OR, Springer, vol. 23(1), pages 1-52, March.
    • Handle: RePEc:spr:aqjoor:v:23:y:2025:i:1:d:10.1007_s10288-025-00585-z
    DOI: 10.1007/s10288-025-00585-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10288-025-00585-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10288-025-00585-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Louis-Martin Rousseau & Michel Gendreau & Gilles Pesant & Filippo Focacci, 2004. "Solving VRPTWs with Constraint Programming Based Column Generation," Annals of Operations Research, Springer, vol. 130(1), pages 199-216, August.
    2. Martin Desrochers & Jacques Desrosiers & Marius Solomon, 1992. "A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows," Operations Research, INFORMS, vol. 40(2), pages 342-354, April.
    3. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
    4. Gschwind, Timo & Bianchessi, Nicola & Irnich, Stefan, 2019. "Stabilized branch-price-and-cut for the commodity-constrained split delivery vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 278(1), pages 91-104.
    5. Jacques Desrosiers & Marco E. Lübbecke, 2005. "A Primer in Column Generation," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 1-32, Springer.
    6. Tilk, Christian & Rothenbächer, Ann-Kathrin & Gschwind, Timo & Irnich, Stefan, 2017. "Asymmetry matters: Dynamic half-way points in bidirectional labeling for solving shortest path problems with resource constraints faster," European Journal of Operational Research, Elsevier, vol. 261(2), pages 530-539.
    7. Cynthia Barnhart & Ellis L. Johnson & George L. Nemhauser & Martin W. P. Savelsbergh & Pamela H. Vance, 1998. "Branch-and-Price: Column Generation for Solving Huge Integer Programs," Operations Research, INFORMS, vol. 46(3), pages 316-329, June.
    8. Briant, Olivier & Cambazard, Hadrien & Cattaruzza, Diego & Catusse, Nicolas & Ladier, Anne-Laure & Ogier, Maxime, 2020. "An efficient and general approach for the joint order batching and picker routing problem," European Journal of Operational Research, Elsevier, vol. 285(2), pages 497-512.
    9. George B. Dantzig & Philip Wolfe, 1960. "Decomposition Principle for Linear Programs," Operations Research, INFORMS, vol. 8(1), pages 101-111, February.
    10. G. Dantzig & R. Fulkerson & S. Johnson, 1954. "Solution of a Large-Scale Traveling-Salesman Problem," Operations Research, INFORMS, vol. 2(4), pages 393-410, November.
    11. Stefan Irnich & Daniel Villeneuve, 2006. "The Shortest-Path Problem with Resource Constraints and k -Cycle Elimination for k (ge) 3," INFORMS Journal on Computing, INFORMS, vol. 18(3), pages 391-406, August.
    12. Roy E. Marsten, 1975. "The Use of the Box Step Method in Discrete Optimization," NBER Working Papers 0086, National Bureau of Economic Research, Inc.
    13. Luciano Costa & Claudio Contardo & Guy Desaulniers, 2019. "Exact Branch-Price-and-Cut Algorithms for Vehicle Routing," Transportation Science, INFORMS, vol. 53(4), pages 946-985, July.
    14. Claudia Bode & Stefan Irnich, 2012. "Cut-First Branch-and-Price-Second for the Capacitated Arc-Routing Problem," Operations Research, INFORMS, vol. 60(5), pages 1167-1182, October.
    15. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    16. Gilbert Laporte & Yves Nobert & Martin Desrochers, 1985. "Optimal Routing under Capacity and Distance Restrictions," Operations Research, INFORMS, vol. 33(5), pages 1050-1073, October.
    17. Gilbert Laporte & Paolo Toth, 2022. "A gap in scientific reporting," 4OR, Springer, vol. 20(1), pages 169-171, March.
    18. G. B. Dantzig & J. H. Ramser, 1959. "The Truck Dispatching Problem," Management Science, INFORMS, vol. 6(1), pages 80-91, October.
    19. Guy Desaulniers, 2010. "Branch-and-Price-and-Cut for the Split-Delivery Vehicle Routing Problem with Time Windows," Operations Research, INFORMS, vol. 58(1), pages 179-192, February.
    20. repec:inm:orijoo:v:4:y:2022:i:1:p:29-44 is not listed on IDEAS
    21. Jaumard, Brigitte & Semet, Frederic & Vovor, Tsevi, 1998. "A generalized linear programming model for nurse scheduling," European Journal of Operational Research, Elsevier, vol. 107(1), pages 1-18, May.
    22. Hintsch, Timo & Irnich, Stefan, 2020. "Exact solution of the soft-clustered vehicle-routing problem," European Journal of Operational Research, Elsevier, vol. 280(1), pages 164-178.
    23. Roberto Baldacci & Aristide Mingozzi & Roberto Roberti, 2011. "New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem," Operations Research, INFORMS, vol. 59(5), pages 1269-1283, October.
    24. Luciano Costa & Claudio Contardo & Guy Desaulniers & Julian Yarkony, 2022. "Stabilized Column Generation Via the Dynamic Separation of Aggregated Rows," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1141-1156, March.
    25. Claudio Contardo & Jean-François Cordeau & Bernard Gendron, 2014. "An Exact Algorithm Based on Cut-and-Column Generation for the Capacitated Location-Routing Problem," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 88-102, February.
    26. Paolo Toth & Daniele Vigo, 2003. "The Granular Tabu Search and Its Application to the Vehicle-Routing Problem," INFORMS Journal on Computing, INFORMS, vol. 15(4), pages 333-346, November.
    27. Moshe Dror, 1994. "Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW," Operations Research, INFORMS, vol. 42(5), pages 977-978, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luciano Costa & Claudio Contardo & Guy Desaulniers, 2019. "Exact Branch-Price-and-Cut Algorithms for Vehicle Routing," Transportation Science, INFORMS, vol. 53(4), pages 946-985, July.
    2. Katrin Heßler & Stefan Irnich, 2023. "Partial Dominance in Branch-Price-and-Cut for the Basic Multicompartment Vehicle-Routing Problem," INFORMS Journal on Computing, INFORMS, vol. 35(1), pages 50-65, January.
    3. Katrin Heßler & Stefan Irnich, 2021. "Partial Dominance in Branch-Price-and-Cut for the Basic Multi-Compartment Vehicle-Routing Problem," Working Papers 2115, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    4. Stefan Faldum & Sarah Machate & Timo Gschwind & Stefan Irnich, 2024. "Partial dominance in branch-price-and-cut algorithms for vehicle routing and scheduling problems with a single-segment tradeoff," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 46(4), pages 1063-1097, December.
    5. Stefan Faldum & Timo Gschwind & Stefan Irnich, 2023. "Subset-Row Inequalities and Unreachability in Path-based Formulations for Routing and Scheduling Problems," Working Papers 2310, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    6. Guy Desaulniers & Diego Pecin & Claudio Contardo, 2019. "Selective pricing in branch-price-and-cut algorithms for vehicle routing," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 8(2), pages 147-168, June.
    7. Dayarian, Iman & Crainic, Teodor Gabriel & Gendreau, Michel & Rei, Walter, 2015. "A column generation approach for a multi-attribute vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 241(3), pages 888-906.
    8. Andrew Allman & Qi Zhang, 2021. "Branch-and-price for a class of nonconvex mixed-integer nonlinear programs," Journal of Global Optimization, Springer, vol. 81(4), pages 861-880, December.
    9. Hoogendoorn, Y.N. & Dalmeijer, K., 2021. "Resource-robust valid inequalities for set covering and set partitioning models," Econometric Institute Research Papers EI 2020-08, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    10. Timo Hintsch & Stefan Irnich, 2018. "Exact Solution of the Soft-Clustered Vehicle Routing Problem," Working Papers 1813, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    11. Omid Shahvari & Rasaratnam Logendran & Madjid Tavana, 2022. "An efficient model-based branch-and-price algorithm for unrelated-parallel machine batching and scheduling problems," Journal of Scheduling, Springer, vol. 25(5), pages 589-621, October.
    12. Jiliu Li & Zhixing Luo & Roberto Baldacci & Hu Qin & Zhou Xu, 2023. "A New Exact Algorithm for Single-Commodity Vehicle Routing with Split Pickups and Deliveries," INFORMS Journal on Computing, INFORMS, vol. 35(1), pages 31-49, January.
    13. Boschetti, Marco Antonio & Maniezzo, Vittorio & Strappaveccia, Francesco, 2017. "Route relaxations on GPU for vehicle routing problems," European Journal of Operational Research, Elsevier, vol. 258(2), pages 456-466.
    14. Guy Desaulniers & Jørgen G. Rakke & Leandro C. Coelho, 2016. "A Branch-Price-and-Cut Algorithm for the Inventory-Routing Problem," Transportation Science, INFORMS, vol. 50(3), pages 1060-1076, August.
    15. Diego Pecin & Claudio Contardo & Guy Desaulniers & Eduardo Uchoa, 2017. "New Enhancements for the Exact Solution of the Vehicle Routing Problem with Time Windows," INFORMS Journal on Computing, INFORMS, vol. 29(3), pages 489-502, August.
    16. Stefan Irnich & Guy Desaulniers & Jacques Desrosiers & Ahmed Hadjar, 2010. "Path-Reduced Costs for Eliminating Arcs in Routing and Scheduling," INFORMS Journal on Computing, INFORMS, vol. 22(2), pages 297-313, May.
    17. Jiachen Zhang & Youcef Magnouche & Pierre Bauguion & Sebastien Martin & J. Christopher Beck, 2024. "Computing Bipath Multicommodity Flows with Constraint Programming–Based Branch-and-Price-and-Cut," INFORMS Journal on Computing, INFORMS, vol. 36(6), pages 1634-1653, December.
    18. Miao Yu & Viswanath Nagarajan & Siqian Shen, 2022. "Improving Column Generation for Vehicle Routing Problems via Random Coloring and Parallelization," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 953-973, March.
    19. Ciancio, Claudio & Laganá, Demetrio & Vocaturo, Francesca, 2018. "Branch-price-and-cut for the Mixed Capacitated General Routing Problem with Time Windows," European Journal of Operational Research, Elsevier, vol. 267(1), pages 187-199.
    20. Yang, Fei & Dai, Ying & Ma, Zu-Jun, 2020. "A cooperative rich vehicle routing problem in the last-mile logistics industry in rural areas," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 141(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:aqjoor:v:23:y:2025:i:1:d:10.1007_s10288-025-00585-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.