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Arc-dependent networks: theoretical insights and a computational study

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Abstract

In this paper, we study the efficacy of several mathematical programming formulations for the single-source shortest path problem, the negative cost cycle detection problem, and the shortest negative cost cycle problem in arc-dependent networks. In an arc-dependent network, the cost of an arc a depends upon the arc preceding a. These networks differ from traditional networks in which the cost associated with an arc is a fixed constant and part of the input. Arc-dependent networks are useful for modeling a number of real-world problems, such as the turn-penalty shortest path problem, which cannot be captured in the traditional network setting. We present new integer and non-linear programming formulations for each problem. We also perform the first known empirical study for arc-dependent networks to contrast the execution times of the two formulations on a set of graphs with varying families and sizes. Our experiments indicate that although non-linear programming formulations are more compact, integer programming formulations are more efficient for the problems studied in this paper. Additionally, we introduce a number of cuts for each integer programming formulation and examine their effectiveness.

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Data availibility

The datasets generated during and/or analyzed during the current study are available in the GitHub repository located at https://github.com/mdwilliamson29/mpc.

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Authors and Affiliations

  1. Department of Computer Science, University of Colorado, Boulder, CO, USA

    Alvaro Velasquez

  2. LDCSEE, West Virginia University, Morgantown, WV, USA

    P. Wojciechowski & K. Subramani

  3. MCS, Marietta College, Marietta, OH, USA

    Matthew Williamson

Authors
  1. Alvaro Velasquez
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  2. P. Wojciechowski
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  3. K. Subramani
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  4. Matthew Williamson
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Contributions

All authors contributed to the work done in this paper. PW and KS designed the integer and non-linear programming formulations and proved that these formulations solve their respective problems in arc-dependent networks. AV performed the empirical study, including implementation and data collection. MW wrote the draft of the manuscript which included formalizing the programming formulations and reporting the results of the empirical study. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to K. Subramani.

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This research was supported in part by the Air-Force Office of Scientific Research through Grant FA9550-19-1-0177. This research was also supported in part by the Air-Force Research Laboratory Rome through Contract FA8750-17-S-7007.

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Velasquez, A., Wojciechowski, P., Subramani, K. et al. Arc-dependent networks: theoretical insights and a computational study. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-05910-z

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