Inverse max $+$ sum spanning tree problem under Hamming distance by modifying the sum-cost vector

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$_0$.