An Optimal Algorithm for Single Maximum Coverage Location on Trees and Related Problems

The single maximum coverage location problem is as follows. We are given an edge-weighted tree with customers located at the nodes. Each node u is associated with a demand w(u) and a radius r(u). The goal is to find, for some facility, a node x such that the total demand of customers u whose distance to x is at most r(u) is maximized. We give a simple O(n log n) algorithm for this problem which improves upon the previously fastest algorithms. We complement this result by an Omega(n log n) lower bound showing that our algorithm is optimal. We observe that our algorithm leads also to improved time bounds for several other location problems such as indirect covering subtree and certain competitive location problems. Finally, we outline how our algorithm can be extended to a large class of distance-based location problems.