Abstract
Finding the shortest path in a directed graph is one of the most important combinatorial optimization problems, having applications in a wide range of fields. In its basic version, however, the problem fails to represent situations in which the value of the objective function is determined not only by the choice of each single arc, but also by the combined presence of pairs of arcs in the solution. In this paper we model these situations as a Quadratic Shortest Path Problem, which calls for the minimization of a quadratic objective function subject to shortest-path constraints. We prove strong NP-hardness of the problem and analyze polynomially solvable special cases, obtained by restricting the distance of arc pairs in the graph that appear jointly in a quadratic monomial of the objective function. Based on this special case and problem structure, we devise fast lower bounding procedures for the general problem and show computationally that they clearly outperform other approaches proposed in the literature in terms of their strength.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Amaldi, E., Galbiati, G., Maffioli, F.: On minimum reload cost paths, tours, and flows. Networks 57(3), 254–260 (2011)
Bellman, R.: On a Routing Problem. Quarterly of Applied Mathematics 16, 87–90 (1958)
Billionnet, A., Elloumi, S., Plateau, M.C.: Improving the performance of standard solvers for quadratic 0–1 programs by a tight convex reformulation: The QCR method. Discrete Applied Mathematics 157(6), 1185–1197 (2009)
Buchheim, C., Traversi, E.: Quadratic 0–1 optimization using separable underestimators. Tech. rep., Optimization Online (2015)
Caprara, A.: Constrained 0–1 quadratic programming: Basic approaches and extensions. European Journal of Operational Research 187(3), 1494–1503 (2008)
Carraresi, P., Malucelli, F.: A new lower bound for the quadratic assignment problem. Operations Research 40(1–Supplement–1), S22–S27 (1992)
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1(1), 269–271 (1959)
Gilmore, P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. Journal of the Society for Industrial & Applied Mathematics 10(2), 305–313 (1962)
Lawler, E.L.: The quadratic assignment problem. Management science 9(4), 586–599 (1963)
Murakami, K., Kim, H.S.: Comparative study on restoration schemes of survivable atm networks. In: INFOCOM 1997. Sixteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings IEEE, vol. 1, pp. 345–352. IEEE (1997)
Nikolova, E., Kelner, J.A., Brand, M., Mitzenmacher, M.: Stochastic shortest paths via quasi-convex maximization. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 552–563. Springer, Heidelberg (2006)
Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23(3), 555–565 (1976)
Sen, S., Pillai, R., Joshi, S., Rathi, A.K.: A mean-variance model for route guidance in advanced traveler information systems. Transportation Science 35(1), 37–49 (2001)
Sivakumar, R.A., Batta, R.: The variance-constrained shortest path problem. Transportation Science 28(4), 309–316 (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Rostami, B., Malucelli, F., Frey, D., Buchheim, C. (2015). On the Quadratic Shortest Path Problem. In: Bampis, E. (eds) Experimental Algorithms. SEA 2015. Lecture Notes in Computer Science(), vol 9125. Springer, Cham. https://doi.org/10.1007/978-3-319-20086-6_29
Download citation
DOI: https://doi.org/10.1007/978-3-319-20086-6_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20085-9
Online ISBN: 978-3-319-20086-6
eBook Packages: Computer ScienceComputer Science (R0)