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.