Abstract
In our modern societies, a certain number of people do not own a car, by choice or by obligation. For some trips, there is no or few alternatives to the car. One way to make these trips possible for these people is to be transported by others who have already planned their trips. We propose to model this problem using as path-finding problem in a list edge-colored graph. This problem is a generalization of the \(s-t\)-path problem, studied by Böhmová et al. We study two optimization functions: minimizing the number of color changes and minimizing the number of colors. We study the complexity and the approximation of this problem. We show the existence of polynomial cases. We show that this problem is NP-complete and hard to approximate, even in restricted cases. Finally, we provide an approximation algorithm.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Böhmová, K., Häfliger, L., Mihalák, M., Pröger, T., Sacomoto, G., Sagot, M.-F.: Computing and listing ST-paths in public transportation networks. Theory Comput. Syst. 62(3), 600–621 (2018)
Broersma, H., Li, X., Woeginger, G.J., Zhang, S.: Paths and cycles in colored graphs. Electron. J. Comb. 31, 299–312 (2005)
Cao, Y., Chen, G., Jing, G., Stiebitz, M., Toft, B.: Graph edge coloring: a survey. Graphs Combin. 35(1), 33–66 (2019)
Captivo, M.E., Clímaco, J.C.N., Pascoal, M.M.B.: A mixed integer linear formulation for the minimum label spanning tree problem. Comput. Oper. Res. 36(11), 3082–3085 (2009)
Chwatal, A.M., Raidl, G.R.: Solving the minimum label spanning tree problem by mathematical programming techniques. Adv. Oper. Res. 2011 (2011)
Cordeau, J.-F., Laporte, G.: The dial-a-ride problem (DARP): variants, modeling issues and algorithms. Q. J. Belg. Fr. Ital. Oper. Res. Soc. 1(2), 89–101 (2003)
Cordeau, J.-F., Laporte, G.: The dial-a-ride problem: models and algorithms. Ann. Oper. Res. 153(1), 29–46 (2007)
Fellows, M., Guo, J., Kanj, I.: The parameterized complexity of some minimum label problems. J. Comput. Syst. Sci. 76(8), 727–740 (2010)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. In: 25th Annual Symposium on Foundations of Computer Science, pp. 338–346 (1984)
Furuhata, M., Dessouky, M., Ordonez, F., Brunet, M.-E., Wang, X., Koenig, S.: Ridesharing: the state-of-the-art and future directions. Transport. Res. B Meth. 57, 28–46 (2013)
Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems. J. Comb. Optim. 14(4), 437–453 (2007)
Ho, S.C., Szeto, W.Y., Kuo, Y.-H., Leung, J.M.Y., Petering, M., Tou, T.W.H.: A survey of dial-a-ride problems: literature review and recent developments. Transport. Res. B Meth. 111, 395–421 (2018)
Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Krumke, S.O., Wirth, H.-C.: On the minimum label spanning tree problem. Inf. Process. Lett. 66(2), 81–85 (1998)
Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the Exponential Time Hypothesis. Bull. EATCS 105, 41–72 (2011)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)
Mourad, A., Puchinger, J., Chu, C.: A survey of models and algorithms for optimizing shared mobility. Transport. Res. B Meth. 123, 323–346 (2019)
Niels, A., Agatz, H., Erera, A.L., Savelsbergh, M.W.P., Wang, X.: Optimization for dynamic ride-sharing: a review. Eur. J. Oper. Res. 223(2), 295–303 (2012)
Woeginger, G.J.: Exact algorithms for NP-hard problems: a survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization — Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36478-1_17
Xiongm, Y., Golden, B., Wasil, E., Chen, S.: The label-constrained minimum spanning tree problem. In: Raghavan, S., Golden, B., Wasil, E. (eds.) Telecommunications Modeling, Policy, and Technology, vol. 44, pp. 39–58. Springer, Boston (2008). https://doi.org/10.1007/978-0-387-77780-1_3
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Davot, T., Giroudeau, R., König, JC. (2021). Complexity and Approximation Results on the Shared Transportation Problem. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-92681-6_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-92680-9
Online ISBN: 978-3-030-92681-6
eBook Packages: Computer ScienceComputer Science (R0)