Abstract
In the field of transportation planning, it is often insufficient to model transportation networks by using networks with fixed arc costs. There may be additional factors that modify the time or cost of a single trip. These include turn prohibitions, fare rebates, and transfer times. Each of these factors causes the cost of a portion of the trip to depend directly on the previous portion of the trip. This dependence can be modeled using arc-dependent networks. In an arc-dependent network, the cost of an arc a depends upon the arc used to enter a. In this paper, we study the approximability of a number of negative cost cycle problems in arc-dependent networks. In a general network, the cost of an arc is a fixed constant and part of the input. Arc-dependent networks can be used to model several real-world problems, including the turn-penalty shortest path problem. Previous literature established that corresponding path problems in these networks are NP-hard. We extend that research by providing inapproximability results for several of these problems. In [7], it was established that a more general form of the shortest path problem in arc-dependent networks, known as the quadratic shortest path problem, cannot be approximated to within a constant factor. In this paper, we strengthen that result by showing NPO PB-completeness.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Bellman, R.E.: Dynamic Programming. Princeton University Press, Princeton (1957)
Caldwell, T.: On finding minimum routes in a network with turn penalties. Commun. ACM 4(2), 107–108 (1961)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)
Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Theory of Parameterized Preprocessing. Cambridge University Press, Kernelization (2019)
Hu, H., Sotirov, R.: Special cases of the quadratic shortest path problem. J. Comb. Optim. 35(3), 754–777 (2017). https://doi.org/10.1007/s10878-017-0219-9
Hao, H., Sotirov, R.: On solving the quadratic shortest path problem. INFORMS J. Comput. 32(2), 219–233 (2020)
Kann, V.: On the approximability of NP-complete Optimization Problems. Ph.D. thesis, Royal Institute of Technology Stockholm (1992)
Kann, V.: Polynomially bounded minimization problems that are hard to approximate. Nordic J. Comput. 1(3), 317–331 (1994)
Orponen, P., Mannila, H.: On approximation preserving reductions: complete problems and robust measures. Technical report, Department of Computer Science, University of Helsinki (1987)
Schöbel, A., Urban, R.: Cheapest paths in public transport: properties and algorithms. In: Huisman, D., Zaroliagis, C.D. (eds.) 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020), vol. 85. OpenAccess Series in Informatics (OASIcs), pp. 13:1–13:16. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2020)
Tan, J., Leong, H.W.: Least-cost path in public transportation systems with fare rebates that are path- and time-dependent. In: Proceedings, The 7th International IEEE Conference on Intelligent Transportation Systems (IEEE Cat. No. 04TH8749), pp. 1000–1005, October 2004
Wojciechowski, P., Williamson, M., Subramani, K.: On finding shortest paths in arc-dependent networks. In: Baïou, M., Gendron, B., Günlük, O., Mahjoub, A.R. (eds.) ISCO 2020. LNCS, vol. 12176, pp. 249–260. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53262-8_21
Yap, C.K.: Some consequences of non-uniform conditions on uniform classes. Theoret. Comput. Sci. 26(3), 287–300 (1983)
Ziliaskopoulos, A.K., Mahmassani, H.S.: A note on least time path computation considering delays and prohibitions for intersection movements. Transp. Res. Part B Methodol. 30(5), 359–367 (1996)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Wojciechowski, P., Subramani, K., Velasquez, A., Williamson, M. (2022). On the Approximability of Path and Cycle Problems in Arc-Dependent Networks. In: Balachandran, N., Inkulu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2022. Lecture Notes in Computer Science(), vol 13179. Springer, Cham. https://doi.org/10.1007/978-3-030-95018-7_23
Download citation
DOI: https://doi.org/10.1007/978-3-030-95018-7_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-95017-0
Online ISBN: 978-3-030-95018-7
eBook Packages: Computer ScienceComputer Science (R0)