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Time-Dependent Shortest Path Problems with Penalties and Limits on Waiting

Author

Listed:
  • Edward He

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • Natashia Boland

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • George Nemhauser

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • Martin Savelsbergh

    (H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

Abstract

Waiting at the right location at the right time can be critically important in certain variants of time-dependent shortest path problems. We investigate the computational complexity of time-dependent shortest path problems in which there is either a penalty on waiting or a limit on the total time spent waiting at a given subset of the nodes. We show that some cases are nondeterministic polynomial-time hard, and others can be solved in polynomial time, depending on the choice of the subset of nodes, on whether waiting is penalized or constrained, and on the magnitude of the penalty/waiting limit parameter. Summary of Contributions: This paper addresses simple yet relevant extensions of a fundamental problem in Operations Research: the Shortest Path Problem (SPP). It considers time-dependent variants of SPP, which can account for changing traffic and/or weather conditions. The first variant that is tackled allows for waiting at certain nodes but at a cost. The second variant instead places a limit on the total waiting. Both variants have applications in transportation, e.g., when it is possible to wait at certain locations if the benefits outweigh the costs. The paper investigates these problems using complexity analysis and algorithm design, both tools from the field of computing. Different cases are considered depending on which of the nodes contribute to the waiting cost or waiting limit (all nodes, all nodes except the origin, a subset of nodes…). The computational complexity of all cases is determined, providing complexity proofs for the variants that are NP-Hard and polynomial time algorithms for the variants that are in P .

Suggested Citation

  • Edward He & Natashia Boland & George Nemhauser & Martin Savelsbergh, 2021. "Time-Dependent Shortest Path Problems with Penalties and Limits on Waiting," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 997-1014, July.
  • Handle: RePEc:inm:orijoc:v:33:y:2021:i:3:p:997-1014
    DOI: 10.1287/ijoc.2020.0985
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    References listed on IDEAS

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    3. Nachtigall, K., 1995. "Time depending shortest-path problems with applications to railway networks," European Journal of Operational Research, Elsevier, vol. 83(1), pages 154-166, May.
    4. Benjamin S Cooper & Raghvendra V Cowlagi, 2018. "Path-planning with waiting in spatiotemporally-varying threat fields," PLOS ONE, Public Library of Science, vol. 13(8), pages 1-21, August.
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