Abstract
This paper focuses on tackling the challenge of determining the longest path on a tree represented as \(T=(V,E)\), where the edges are assigned interval lengths. We employ the minmax regret concept as a metric to quantify robustness. The longest path associated with this concept is referred to as the minmax regret longest path on the tree T. Initially, we establish certain structural properties pertaining to both the longest and second longest paths within the tree. Then we reset the problem as a formulation regarding the longest and second longest path w.r.t candidate scenarios. We finally devise a combinatorial algorithm capable of identifying the minmax regret longest path within the tree in \(O(|\mathcal L|^2|V|)\) time with \(\mathcal L\) being the set of leaf nodes.
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References
Bachtler, O., Krumke, S.O., Le, H.M.: Robust single machine makespan scheduling with release date uncertainty. Oper. Res. Lett. 48, 816–819 (2020)
Ben-Tal, A., Nemirovski, A., El-Ghaoui, L.: Robust optimization. Princeton University Press, Princeton, New Jersey (2009)
Björklund, A., Husfeldt, T., Khanna, S.: Approximating longest directed paths and cycles. In: Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP 2004), Lecture Notes in Computer Science, vol. 3142, pp. 222–233. Springer-Verlag, Berlin (2004)
Burkard, R.E., Dollani, H.: Robust location problems with pos/neg weights on a tree. Networks 38, 102–113 (2001)
Chassein, A., Goerigk, M.: Variable-sized uncertainty and inverse problems in robust optimization. Eur. J. Oper. Res. 264, 17–28 (2018)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP Completeness. Freeman, San Francisco, CA (1979)
Gutin, G.: Finding a longest path in a complete multipartite digraphs. SIAM J. Disc. Math. 6(2), 270–273 (1993)
Handler, G.Y.: Minimax location of a facility in an undirected tree networks. Transp. Sci. 7, 287–293 (1973)
Karger, D., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18, 82–98 (1997)
Kasperski, A., Makuchowski, M., Zieliński, P.: A Tabu search algorithm for the minimax regret minimum spanning tree problem with interval data. J. Heuristics 18(4), 593–625 (2012)
Kouvelis, P., Yu, G.: Robust Discrete Optimization and Its Applications. Kluwer Academic Publishers (1997)
Monien, B.: How to find long paths efficiently. Ann. Discrete Math. 25, 239–254 (1985)
Montemanni, R., Gambardella, L.M.: An exact algorithm for the robust shortest path problem with interval data. Comput. Oper. Res. 31, 1667–1680 (2004)
Montemanni, R., Gambardella, L.M., Donati, A.V.: A branch and bound algorithm for the robust shortest path problem with interval data. Oper. Res. Lett. 32, 225–232 (2004)
Nguyen, K.T., Hung, N.T.: The minmax regret inverse maximum weight problem. Appl. Math. Comput. 407, 126328 (2021)
Pascoal, M.M.B., Resende, M.: The minmax regret robust shortest path problem in a finite multi-scenario model. Appl. Math. Comput. 241, 88–111 (2014)
Pérez-Galarce, F., Candia-Vejar, A., Astudillo, C.A., Bardeen, M.D.: Algorithms for the minmax regret path problem with interval data. Inf. Sci. 462, 218–241 (2018)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer-Verlag, Berlin Heidelberg (2003)
Williams, R.: Finding paths of length \(k\) in \(O^*(2^k)\) time. Inf. Process. Lett. 109, 315–318 (2009)
Ye, J.H., Wang, B.F.: On the minmax regret path median problem on trees. J. Comput. Syst. Sci. 81(7), 1159–1170 (2015)
Yu, G., Yang, J.: On the robust shortest path problem. Comput. Oper. Res. 25(6), 457–468 (1998)
Zieliński, P.: The computational complexity of the relative robust shortest path problem with interval data. Eur. J. Oper. Res. 158, 570–576 (2004)
Acknowlegement
The authors thank the anonymous referees for valuable comments that helped to improved the paper. The fourth author (H.M. Le) would like to thank Van Lang University, Vietnam, for funding this work.
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Quy, T.Q., Ly, N.H.C., Cuong, D.D., Le, H.M. (2023). The Minmax Regret Longest Path of a Tree with Interval Edge Lengths. In: Nghia, P.T., Thai, V.D., Thuy, N.T., Son, L.H., Huynh, VN. (eds) Advances in Information and Communication Technology. ICTA 2023. Lecture Notes in Networks and Systems, vol 847. Springer, Cham. https://doi.org/10.1007/978-3-031-49529-8_35
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DOI: https://doi.org/10.1007/978-3-031-49529-8_35
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