An FPTAS for generalized absolute 1-center problem in vertex-weighted graphs

Abstract

Given a vertex-weighted undirected connected graph $G = (V, E, \ell , \rho )$, where each edge $e \in E$ has a length $\ell (e) > 0$ and each vertex $v \in V$ has a weight $\rho (v) > 0$, a subset $T \subseteq V$ of vertices and a set S containing all the points on edges in a subset $E' \subseteq E$ of edges, the generalized absolute 1-center problem (GA1CP), an extension of the classic vertex-weighted absolute 1-center problem (A1CP), asks to find a point from S such that the longest weighted shortest path distance in G from it to T is minimized. This paper presents a simple FPTAS for GA1CP by traversing the edges in $E'$ using a positive real number as step size. The FPTAS takes $O( |E| |V| + |V|^2 \log \log |V| + \frac{1}{\epsilon } |E'| |T| {\mathcal {R}})$ time, where ${\mathcal {R}}$ is an input parameter size of the problem instance, for any given $\epsilon > 0$. For instances with a small input parameter size ${\mathcal {R}}$, applying the FPTAS with $\epsilon = \Theta (1)$ to the classic vertex-weighted A1CP can produce a $(1 + \Theta (1))$-approximation in at most O(|E| |V|) time when the distance matrix is known and $O(|E| |V| + |V|^2 \log \log |V|)$ time when the distance matrix is unknown, which are smaller than Kariv and Hakimi’s $O(|E| |V| \log |V|)$-time algorithm and $O(|E| |V| \log |V| + |V|^3)$-time algorithm, respectively.