Computation of Inverse 1-Center Location Problem on the Weighted Trapezoid Graphs

Abstract

Let $T_{TRP}$ be the tree corresponding to the weighted trapezoid graph $G=(V,E)$. The eccentricity $e(v)$ of the vertex $v$ is defined as the sum of the weights of the vertices from $v$ to the vertex farthest from $v \in T_{TRP}$. A vertex with minimum eccentricity in the tree $T_{TRP}$ is called the 1-center of that tree. In an inverse 1-center location problem, the parameter of the tree $T_{TRP}$ corresponding to the weighted trapezoid graph $G=(V,E)$, like vertex weights, have to be modified at minimum total cost such that a pre-specified vertex $s \in V$ becomes the 1-center of the trapezoid graph $G$. In this paper, we present an optimal algorithm to find an inverse 1-center location on the weighted tree $T_{TRP}$ corresponding to the weighted trapezoid graph $G=(V,E)$, where the vertex weights can be changed within certain bounds. The time complexity of our proposed algorithm is $O(n)$, where $n$ is the number of vertices of the trapezoid graph $G$.