# Inverse max \(+\) sum spanning tree problem under Hamming distance by modifying the sum-cost vector

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## Abstract

The inverse max \(+\) sum spanning tree (MSST) problem is considered by modifying the sum-cost vector under the Hamming Distance. On an undirected network *G*(*V*, *E*, *w*, *c*), a weight *w*(*e*) and a cost *c*(*e*) are prescribed for each edge \(e\in E\). The MSST problem is to find a spanning tree \(T^*\) which makes the combined weight \(\max _{e\in T}w(e)+\sum _{e\in T}c(e)\) as small as possible. It can be solved in \(O(m\log n)\) time, where \(m:=|E|\) and \(n:=|V|\). Whereas, in an inverse MSST problem, a given spanning tree \(T_0\) of *G* is not an optimal MSST. The sum-cost vector *c* is to be modified to \(\bar{c}\) so that \(T_0\) becomes an optimal MSST of the new network \(G(V,E,w,\bar{c})\) and the cost \(\Vert \bar{c}-c\Vert \) can be minimized under Hamming Distance. First, we present a mathematical model for the inverse MSST problem and a method to check the feasibility. Then, under the weighted bottleneck-type Hamming distance, we design a binary search algorithm whose time complexity is \(O(m log^2 n)\). Next, under the unit sum-type Hamming distance, which is also called \(l_0\) norm, we show that the inverse MSST problem (denoted by IMSST\(_0\)) is \(NP-\)hard. Assuming \({\textit{NP}} \nsubseteq {\textit{DTIME}}(m^{{\textit{poly}} \log m})\), the problem IMSST\(_0\) is not approximable within a factor of \(2^{\log ^{1-\varepsilon } m}\), for any \(\varepsilon >0\). Finally, We consider the augmented problem of IMSST\(_0\) (denoted by AIMSST\(_0\)), whose objective function is to multiply the \(l_0\) norm \(\Vert \beta \Vert _0\) by a sufficiently large number *M* plus the \(l_1\) norm \(\Vert \beta \Vert _1\). We show that the augmented problem and the \(l_1\) norm problem have the same Lagrange dual problems. Therefore, the \(l_1\) norm problem is the best convex relaxation (in terms of Lagrangian duality) of the augmented problem AIMSST\(_0\), which has the same optimal solution as that of the inverse problem IMSST\(_0\).

### Keywords

Max \(+\) sum spanning tree problem Inverse optimization problem Hamming distance \(l_0\) norm Approximability## Notes

### Acknowledgements

Work of P.M. Pardalos was conducted at the National Research University Higher School of Economics and supported by RSF grant 14-41-00039.

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