Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted $l_\infty$ Norm

Abstract

The inverse max + sum spanning tree (IMSST) problem is studied, which is the first inverse problem on optimization problems with combined minmax–minsum objective functions. Given an edge-weighted undirected network $G(V,E,c,w)$, the MSST problem is to find a spanning tree $T$ which minimizes the combined weight $\max _{e\in T}w(e)+\sum _{e\in T}c(e)$, which can be solved in $O(m\log n)$ time, where $m:=|E|$ and $n:=|V|$. Whereas, in an IMSST problem, a spanning tree $T_0$ of $G$ is given, which is not an optimal MSST. A new sum-cost vector $\bar{c}$ is to be identified so that $T_0$ becomes an optimal MSST of the network $G(V,E,\bar{c},w)$, where $0\le c-l\le \bar{c} \le c+u$ and $l,u\ge 0$. The objective is to minimize the cost $\max _{e\in E}q(e)|\bar{c}(e)-c(e)|$ incurred by modifying the sum-cost vector $c$ under weighted $l_\infty$ norm, where $q(e)\ge 1$. We show that the unbounded IMSST problem is a linear fractional combinatorial optimization (LFCO) problem and develop a discrete type Newton method to solve it. Furthermore, we prove an $O(m)$ bound on the number of iterations, although most LFCO problems can be solved in $O(m^2 \log m)$ iterations. Therefore, both the unbounded and bounded IMSST problems can be solved by solving $O(m)$ MSST problems. Computational results show that the algorithms can efficiently solve the problems.