## Abstract

The max+sum spanning tree (**MSST**) problem is to determine a spanning tree *T* whose combined weight \(\max _{e\in T}w(e)+\sum _{e\in T}c(e)\) is minimum for a given edge-weighted undirected network *G*(*V*, *E*, *c*, *w*). This problem can be solved within \(O(m \log n)\) time, where *m* and *n* are the numbers of edges and nodes, respectively. An inverse **MSST** problem (**IMSST**) aims to determine a new weight vector \(\bar{w}\) so that a given \(T^0\) becomes an optimal **MSST** of the new network \(G(V,E,c,\bar{w})\). The **IMSST** problem under weighted \(l_\infty \) norm is to minimize the modification cost \(\max _{e\in E} q(e)|\bar{w}(e)-w(e)|\), where *q*(*e*) is a cost modifying the weight *w*(*e*). We first provide some optimality conditions of the problem. Then we propose a strongly polynomial time algorithm in \(O(m^2\log n)\) time based on a binary search and a greedy method. There are *O*(*m*) iterations and we solve an **MSST** problem under a new weight in each iteration. Finally, we perform some numerical experiments to show the efficiency of the algorithm.

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## References

Ahuja, R.K., Orlin, J.B.: A faster algorithm for the inverse spanning tree problem. J. Algorithm

**34**, 177–193 (2000)Cai, M.-C., Duin, C.W., Yang, X., Zhang, J.: The partial inverse minimum spanning tree problem when weight increasing is forbidden. Eur. J. Oper. Res.

**188**, 348–353 (2008)Duin, C.W., Volgenant, A.: Some inverse optimization problems under the Hamming distance. Eur. J. Oper. Res.

**170**, 887–899 (2006)Guan, X.C., He, X.Y., Pardalos, P.M., Zhang, B.W.: Inverse max + sum spanning tree problem under Hamming distance by modifying the sum-cost vector. J. Glob. Optim.

**69**, 911–925 (2017)Guan, X.C., Pardalos, P.M., Zuo, X.: Inverse max + sum spanning tree problem by modifying the sum-cost vector under weighted \(l_{\infty }\) Norm. J. Glob. Optim.

**61**(1), 165–182 (2015)Guan, X.C., Pardalos, P.M., Zhang, B.W.: Inverse max+sum spanning tree problem under weighted \(l_1\) norm by modifying the sum-cost vector. Optim. Lett.

**5**, 1–13 (2017)Guan, X.C., Zhang, B.W.: Inverse 1-median problem on trees under weighted Hamming distance. J. Glob. Optim.

**54**(1), 75–82 (2012)Guan, X.C., Zhang, J.Z.: Inverse constrained bottleneck problems under weighted \(l_{\infty }\) norm. Comput. Oper. Res.

**34**, 3243–3254 (2007)Heuburger, C.: Inverse optimization, a survey on problems, methods, and results. J. Comb. Optim.

**8**(3), 329–361 (2004)Hochbaum, D.S.: Efficient algorithms for the inverse spanning tree problem. Oper. Res.

**51**(5), 785–797 (2003)He, Y., Zhang, B., Yao, E.: Weighted inverse minimum spanning tree problems under Hamming distance. J. Comb. Optim.

**9**, 91–100 (2005)Lai, T., Orlin, J.: The Complexity of Preprocessing. Research Report of Sloan School of Management. MIT, Cambridge (2003)

Li, X., Shu, X., Huang, H., Bai, J.: Capacitated partial inverse maximum spanning tree under the weighted Hamming distance. J. Comb. Optim.

**38**(4), 1005–1018 (2019)Liu, L.C., Wang, Q.: Constrained inverse min-max spanning tree problems under the weighted Hamming distance. J. Glob. Optim.

**43**, 83–95 (2009)Li, X., Zhang, Z., Du, D.-Z.: Partial inverse maximum spanning tree in which weight can only be decreased under \(l_p\) norm. J. Glob. Optim.

**30**, 677–685 (2018)Li, S., Zhang, Z., Lai, H.-J.: Algorithms for constraint partial inverse matroid problem with weight increase forbidden. Theor. Comput. Sci.

**640**, 119–124 (2016)Li, X., Zhang, Z., Yang, R., Zhang, H., Du, D.-Z.: Approximation algorithms for capacitated partial inverse maximum spanning tree problem. J. Glob. Optim.

**77**(2), 319–340 (2020)Minoux, M.: Solving combinatorial problemswith combined minmax-minsum objective and applications. Math. Program.

**45**, 361–371 (1989)Punnen, A.P.: On combined minmax-minsum optimization. Comput. Oper. Res.

**21**(6), 707–716 (1994)Punnen, A.P., Nair, K.P.K.: An \(O(m\log n)\) algorithm for the max + sum spanning tree problem. Eur. J. Oper. Res.

**89**, 423–426 (1996)Sokkalingam, P.T., Ahuja, R.K., Orlin, J.B.: Solving inverse spanning tree problems through network flow techniques. Oper. Res.

**47**(2), 291–298 (1999)Tayyebi, J., Sepasian, A.R.: Partial inverse min-max spanning tree problem. J. Comb. Optim.

**40**(4), 1075–1091 (2020)Yang, X.G., Zhang, J.Z.: Some inverse min-max network problems under weighted \(l_1\) and \(l_{\infty }\) norms with bound constraints on changes. J. Comb. Optim.

**13**, 123–135 (2007)Zhang, J.Z., Liu, Z.: A general model of some inverse combinatorial optimization problems and its solution method under \(l_{\infty }\) norm. J. Comb. Optim.

**6**, 207–227 (2002)Zhang, B., Zhang, J., He, Y.: Constrained inverse minimum spanning tree problems under the bottlenecktype Hamming distance. J. Glob. Optim.

**34**(3), 467–474 (2006)

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Research is supported by National Natural Science Foundation of China (11471073).

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Jia, J., Guan, X., Zhang, Q. *et al.* Inverse max+sum spanning tree problem under weighted \(l_{\infty }\) norm by modifying max-weight vector.
*J Glob Optim* **84**, 715–738 (2022). https://doi.org/10.1007/s10898-022-01170-y

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DOI: https://doi.org/10.1007/s10898-022-01170-y