Efficient Algorithms for Centers and Medians in Interval and Circular-Arc Graphs

The p-center problem is to locate p facilities on a network so as to minimize the largest distance from a demand point to its nearest facility. The p-median problem is to locate p facilities on a network so as to minimize the average distance from a demand point to its closest facility. We consider these problems when the network can be modeled by an interval or circular-arc graph whose edges have unit lengths. We provide, given the interval model of an n vertex interval graph, an O(n) time algorithm for the 1-median problem on the interval graph. We also show how to solve the p-median problem, for arbitrary p, on an interval graph in O(pn log n) time and on a circular-arc graph in O(pn² log n) time. We introduce a spring representation of the objective function and show how to solve the p-center problem on a circular-arc graph in O(pn) time, assuming that the arc endpoints are sorted.