Abstract
An instance of the NP-hard Quadratic Shortest Path Problem (QSPP) is called linearizable iff it is equivalent to an instance of the classic Shortest Path Problem (SPP) on the same input digraph. The linearization problem for the QSPP (LinQSPP) decides whether a given QSPP instance is linearizable and determines the corresponding SPP instance in the positive case. We provide a novel linear time algorithm for the LinQSPP on acyclic digraphs which runs considerably faster than the previously best algorithm. The algorithm is based on a new insight revealing that the linearizability of the QSPP for acyclic digraphs can be seen as a local property. Our approach extends to the more general higher-order shortest path problem.
G. J. Woeginger—Deceased in April 2022.
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Notes
- 1.
We use the same notation for the path P and the set of its arcs.
- 2.
We assume that f is given as an oracle.
References
Bookhold, I.: A contribution to quadratic assignment problems. Optimization 21(6), 933–943 (1990)
Çela, E., Klinz, B., Lendl, S., Orlin, J.B., Woeginger, G.J., Wulf, L.: Linearizable special cases of the quadratic shortest path problem. In: Kowalik, Ł., Pilipczuk, M., Rza̧żewski, P. (eds.) WG 2021. LNCS, vol. 12911, pp. 245–256. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-86838-3_19
Cela, E., Deineko, V.G., Woeginger, G.J.: Linearizable special cases of the QAP. J. Comb. Optim. 31(3), 1269–1279 (2016)
Ćustić, A., Punnen, A.P.: A characterization of linearizable instances of the quadratic minimum spanning tree problem. J. Comb. Optim. 35(2), 436–453 (2018)
De Meijer, F., Sotirov, R.: The quadratic cycle cover problem: special cases and efficient bounds. J. Comb. Optim. 39(4), 1096–1128 (2020)
Erdoğan, G.: Quadratic assignment problem: linearizations and polynomial time solvable cases, Ph. D. thesis, Bilkent University (2006)
Erdoğan, G., Tansel, B.: A branch-and-cut algorithm for quadratic assignment problems based on linearizations. Comput. Oper. Res. 34(4), 1085–1106 (2007)
Erdoğan, G., Tansel, B.C.: Two classes of quadratic assignment problems that are solvable as linear assignment problems. Discret. Optim. 8(3), 446–451 (2011)
Gamvros, I.: Satellite network design, optimization and management. University of Maryland, College Park (2006)
Hu, H., Sotirov, R.: Special cases of the quadratic shortest path problem. J. Comb. Optim. 35(3), 754–777 (2018)
Hu, H., Sotirov, R.: On solving the quadratic shortest path problem. INFORMS J. Comput. 32(2), 219–233 (2020)
Hu, H., Sotirov, R.: The linearization problem of a binary quadratic problem and its applications. Annal. Oper. Res. 307, 229–249 (2021)
Kabadi, S.N., Punnen, A.P.: An \(O(n^4)\) algorithm for the QAP linearization problem. Math. Oper. Res. 36(4), 754–761 (2011)
Murakami, K., Kim, H.S.: Comparative study on restoration schemes of survivable ATM networks. In: Proceedings of INFOCOM1997, vol. 1, pp. 345–352. IEEE (1997)
Nie, Y.M., Wu, X.: Reliable a priori shortest path problem with limited spatial and temporal dependencies. In: Lam, W., Wong, S., Lo, H. (eds.) Transportation and Traffic Theory 2009: Golden Jubilee, pp. 169–195. Springer, Boston (2009). https://doi.org/10.1007/978-1-4419-0820-9_9
Punnen, A.P., Kabadi, S.N.: A linear time algorithm for the Koopmans-Beckmann QAP linearization and related problems. Discret. Optim. 10(3), 200–209 (2013)
Punnen, A.P., Walter, M., Woods, B.D.: A characterization of linearizable instances of the quadratic traveling salesman problem. arXiv preprint arXiv:1708.07217 (2017)
Rostami, B., et al.: The quadratic shortest path problem: complexity, approximability, and solution methods. Eur. J. Oper. Res. 268(2), 473–485 (2018)
Rostami, B., Malucelli, F., Frey, D., Buchheim, C.: On the quadratic shortest path problem. In: Bampis, E. (ed.) SEA 2015. LNCS, vol. 9125, pp. 379–390. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20086-6_29
Sen, S., Pillai, R., Joshi, S., Rathi, A.K.: A mean-variance model for route guidance in advanced traveler information systems. Transp. Sci. 35(1), 37–49 (2001)
Sivakumar, R.A., Batta, R.: The variance-constrained shortest path problem. Transp. Sci. 28(4), 309–316 (1994)
Sotirov, R., Verchére, M.: The quadratic minimum spanning tree problem: lower bounds via extended formulations. arXiv preprint arXiv:2102.10647 (2021)
Waddell, L., Adams, W.: Characterizing linearizable QAPs by the level-1 reformulation-linearization technique. (2021). https://optimization-online.org/?p=17020, preprint
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This research has been supported by the Austrian Science Fund (FWF): W1230.
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Çela, E., Klinz, B., Lendl, S., Woeginger, G.J., Wulf, L. (2023). A Linear Time Algorithm for Linearizing Quadratic and Higher-Order Shortest Path Problems. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_33
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