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On Solving the Quadratic Shortest Path Problem

Author

Listed:
  • Hao Hu

    (CentER, Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, Netherlands)

  • Renata Sotirov

    (Department of Econometrics and Operations Research, Tilburg University, 5000 LE Tilburg, Netherlands)

Abstract

The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order m + 1, where m is the number of arcs in the graph. We use the alternating direction method of multipliers to solve the semidefinite programming relaxations. Numerical results show that our bounds are currently the strongest bounds for the quadratic shortest path problem. We also present computational results on solving the quadratic shortest path problem using a branch and bound algorithm. Our algorithm computes a semidefinite programming bound in each node of the search tree, and solves instances with up to 1,300 arcs in less than an hour.

Suggested Citation

  • Hao Hu & Renata Sotirov, 2020. "On Solving the Quadratic Shortest Path Problem," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 219-233, April.
    • Handle: RePEc:inm:orijoc:v:32:y:2020:i:2:p:219-233
    DOI: 10.1287/ijoc.2018.0861
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    References listed on IDEAS

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    1. Raj A. Sivakumar & Rajan Batta, 1994. "The Variance-Constrained Shortest Path Problem," Transportation Science, INFORMS, vol. 28(4), pages 309-316, November.
    2. Rostami, Borzou & Chassein, André & Hopf, Michael & Frey, Davide & Buchheim, Christoph & Malucelli, Federico & Goerigk, Marc, 2018. "The quadratic shortest path problem: complexity, approximability, and solution methods," European Journal of Operational Research, Elsevier, vol. 268(2), pages 473-485.
    3. Yu Marco Nie & Xing Wu, 2009. "Reliable a Priori Shortest Path Problem with Limited Spatial and Temporal Dependencies," Springer Books, in: William H. K. Lam & S. C. Wong & Hong K. Lo (ed.), Transportation and Traffic Theory 2009: Golden Jubilee, chapter 0, pages 169-195, Springer.
    4. Suvrajeet Sen & Rekha Pillai & Shirish Joshi & Ajay K. Rathi, 2001. "A Mean-Variance Model for Route Guidance in Advanced Traveler Information Systems," Transportation Science, INFORMS, vol. 35(1), pages 37-49, February.
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    Citations

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    Cited by:

    1. Hu, Hao & Sotirov, Renata & Wolkowicz, Henry, 2023. "Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs," Other publications TiSEM 8dd3dbae-58fd-4238-b786-e, Tilburg University, School of Economics and Management.
    2. Alvaro Velasquez & P. Wojciechowski & K. Subramani & Matthew Williamson, 2024. "Arc-dependent networks: theoretical insights and a computational study," Annals of Operations Research, Springer, vol. 338(2), pages 1101-1126, July.
    3. Frank de Meijer & Renata Sotirov, 2021. "SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1262-1276, October.
    4. Hao Hu & Renata Sotirov, 2021. "The linearization problem of a binary quadratic problem and its applications," Annals of Operations Research, Springer, vol. 307(1), pages 229-249, December.
    5. Adrien Durand & Timothé Watteau & Georges Ghazi & Ruxandra Mihaela Botez, 2024. "Generalized Shortest Path Problem: An Innovative Approach for Non-Additive Problems in Conditional Weighted Graphs," Mathematics, MDPI, vol. 12(19), pages 1-24, September.

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