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Solving Elementary Shortest-Path Problems as Mixed-Integer Programs

Author

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  • Michael Drexl

    (Johannes Gutenberg University Mainz)

  • Stefan Irnich

    (Johannes Gutenberg University Mainz)

Abstract

Ibrahim, Maculan, and Minoux (International Transactions in Operational Research, vol. 16, 2009, pp. 361-369) presented and analyzed two integer programming formulations for the elementary shortest-path problem (ESPP), which is known to be NP-hard if the underlying digraph contains negative cycles. In fact, the authors showed that a formulation based on commodity flows possesses a significantly stronger LP-relaxation than a formulation based on arc flow variables. Since the ESPP is essentially an integer problem, the contribution of our paper lies in extending this research by comparing the formulations with regard to the computation time and memory requirements required for their integer solution. Moreover, we assess the quality of the lower bounds provided by an integer relaxation of the commodity flow formulation.

Suggested Citation

  • Michael Drexl & Stefan Irnich, 2012. "Solving Elementary Shortest-Path Problems as Mixed-Integer Programs," Working Papers 1201, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    • Handle: RePEc:jgu:wpaper:1201
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    File URL: https://download.uni-mainz.de/RePEc/pdf/Discussion_Paper_1201.pdf
    File Function: First version, 2012
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    References listed on IDEAS

    as
    1. Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), 2005. "Column Generation," Springer Books, Springer, number 978-0-387-25486-9, December.
    2. Teodor Gabriel Crainic & Nicoletta Ricciardi & Giovanni Storchi, 2009. "Models for Evaluating and Planning City Logistics Systems," Transportation Science, INFORMS, vol. 43(4), pages 432-454, November.
    3. Stefan Irnich & Guy Desaulniers, 2005. "Shortest Path Problems with Resource Constraints," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 33-65, Springer.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Range, Troels Martin, 2013. "Exploiting Set-Based Structures to Accelerate Dynamic Programming Algorithms for the Elementary Shortest Path Problem with Resource Constraints," Discussion Papers on Economics 17/2013, University of Southern Denmark, Department of Economics.
    2. Quoc Trung Bui & Yves Deville & Quang Dung Pham, 2016. "Exact methods for solving the elementary shortest and longest path problems," Annals of Operations Research, Springer, vol. 244(2), pages 313-348, September.
    3. 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.

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