Abstract
We study in this paper the problem of finding in a graph a subset of k edges whose deletion causes the largest increase in the weight of a minimum spanning tree. We propose for this problem an explicit enumeration algorithm whose complexity, when compared to the current best algorithm, is better for general k but very slightly worse for fixed k. More interestingly, unlike in the previous algorithms, we can easily adapt our algorithm so as to transform it into an implicit exploration algorithm based on a branch and bound scheme. We also propose a mixed integer programming formulation for this problem. Computational results show a clear superiority of the implicit enumeration algorithm both over the explicit enumeration algorithm and the mixed integer program.
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References
Bar-Noy, A., Khuller, S., Schieber, B.: The complexity of finding most vital arcs and nodes. Technical Report CS-TR-3539, University of Maryland (1995)
Chazelle, B.: A minimum spanning tree algorithm with inverse-Ackermann type complexity. Journal of the ACM 47(6), 1028–1047 (2000)
Dixon, B., Rauch, M., Tarjan, R.E.: Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Journal on Computing 21(6), 1184–1192 (1992)
Frederickson, G.N., Solis-Oba, R.: Increasing the weight of minimum spanning trees. In: Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms (SODA 1996), pp. 539–546 (1996)
Hsu, L., Jan, R., Lee, Y., Hung, C., Chern, M.: Finding the most vital edge with respect to minimum spanning tree in a weighted graph. Information Processing Letters 39(5), 277–281 (1991)
Iwano, K., Katoh, N.: Efficient algorithms for finding the most vital edge of a minimum spanning tree. Information Processing Letters 48(5), 211–213 (1993)
Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Rudolf, G., Zhao, J.: On short paths interdiction problems: total and node-wise limited interdiction. Theory of Computing Systems 43(2), 204–233 (2008)
Liang, W.: Finding the k most vital edges with respect to minimum spanning trees for fixed k. Discrete Applied Mathematics 113(2-3), 319–327 (2001)
Liang, W., Shen, X.: Finding the k most vital edges in the minimum spanning tree problem. Parallel Computer 23(3), 1889–1907 (1997)
Magnanti, T.L., Wolsey, L.: Optimal trees. In: Ball, M.O., et al. (eds.) Network Models. Handbook in Operations Research and Management Science, vol. 7, pp. 503–615. North-Holland, Amsterdam (1995)
Nardelli, E., Proietti, G., Widmayer, P.: A faster computation of the most vital edge of a shortest path. Information Processing Letters 79(2), 81–85 (2001)
Pettie, S.: Sensitivity analysis of minimum spanning tree in sub-inverse-ackermann time. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 964–973. Springer, Heidelberg (2005)
Pettie, S., Ramachandran, V.: An optimal minimum spanning tree algorithm. Journal of the ACM 49(1), 16–34 (2002)
Ratliff, H.D., Sicilia, G.T., Lubore, S.H.: Finding the n most vital links in flow networks. Management Science 21(5), 531–539 (1975)
Shen, H.: Finding the k most vital edges with respect to minimum spanning tree. Acta Informatica 36(5), 405–424 (1999)
Suraweera, F., Maheshwari, P., Bhattacharya, P.: Optimal algorithms to find the most vital edge of a minimum spanning tree. Technical Report CIT-95-21, School of Computing and Information Technology, Griffith University (1995)
Tarjan, R.E.: Applications of path compression on balanced trees. Journal of the ACM 26(4), 690–715 (1979)
Wollmer, R.: Removing arcs from a network. Operations Research 12(6), 934–940 (1964)
Wood, R.K.: Deterministic network interdiction. Mathematical and Computer Modeling 17(2), 1–18 (1993)
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Bazgan, C., Toubaline, S., Vanderpooten, D. (2011). Efficient Algorithms for Finding the k Most Vital Edges for the Minimum Spanning Tree Problem. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_11
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DOI: https://doi.org/10.1007/978-3-642-22616-8_11
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