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 one of the n demand points to one of the p facilities. 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 an circular-arc graph in O(pn2 log n) time. other than for trees, no polynomial time algorithm for p-median problem has been reported for any large class of graphs. We introduce a spring model of computation and show how to solve the p-center problem on an circular-arc graph in O(pn) time, assuming that the arc endpoints are sorted.