Maximum-Cover Source-Location Problems

Given a graph G = (V, E), a set of vertices S ⊆ V covers v ∈ V if the edge connectivity between S and v is at least a given number k. Vertices in S are called sources. The source location problem is a problem of finding a minimum-size source set covering all vertices of a given graph. This paper presents a new variation of the problem, called maximum-cover source-location problem, which finds a source set S with a given size p, maximizing the sum of the weight of vertices covered by S . It presents an O(np + m + n log n)-time algorithm for k = 2, where n = |V | and m = |E|. Especially it runs linear time if G is connected. This algorithm uses a subroutine for finding a subtree with the maximum weight among p-leaf trees of a given vertex-weighted tree. For the problem we give a greedy-based linear-time algorithm, which is an extension of the linear-time algorithm for finding a longest path of a given tree presented by E. W. Dijkstra around 1960. Moreover, we show some polynomial solvable cases, e.g., a given graph is a tree or (k−1)-edge-connected, and NP-hard cases, e.g., a vertexcost function is given or G is a digraph. key words: graph, edge-connectivity, location problem, polynomial-time algorithm, NP-hard