Approximating the Restricted 1-Center in Graphs

Abstract

This paper studies the restricted vertex 1-center problem (RV1CP) and restricted absolute 1-center problem (RA1CP) in general undirected graphs with each edge having two weights, cost and delay. First, we devise a simple FPTAS for RV1CP with $O(mn^3(\frac{1}{\epsilon }+\log \log n))$ running time, based on FPTAS proposed by Lorenz and Raz (Oper. Res. Lett. 28(1999), 213–219) for computing end-to-end restricted shortest path (RSP). During the computation of the FPTAS for RV1CP, we derive a RSP distance matrix. Next, we discuss RA1CP in such graphs where the delay is a separable (e.g., linear) function of the cost on edge. We investigate an important property that the FPTAS for RV1CP can find a $(1+\epsilon )$-approximation of RA1CP when the RSP distance matrix has a saddle point. In addition, we show that it is harder to find an approximation of RA1CP when the matrix has no saddle point. This paper develops a scaling algorithm with at most $O(mn^3K(\frac{\log K}{\eta }+\log \log n))$ running time where K is a step-size parameter and $\eta$ is a given positive number, to find a $(1+\eta )$-approximation of RA1CP.