Block-vertex duality and the one-median problem

The w-centroid problem, denoted by (C), is an optimization problem which has been shown by Kariv and Hakimi to be equivalent, on a tree graph, to the 1-median location problem, denoted by (M). For a general (weighted) connected graph G we develop a duality between (C) (which is defined on G) and a block optimization problem, denoted by (B), and defined over the blocks of G. A block is a maximal nonseparable subgraph. We analyze (B) and (C) by means of two problems equivalent to (B) and (C) respectively, but defined on a blocking graph G which is always a tree. We give an O(∣V∣) algorithm to solve the two problems on G, and we characterize the solutions. We also show that the solution to a 1-median problem defined on G either solves (M) on the original graph G or localizes the search for a solution to (M) to the vertices of a single block. We introduce an extended version of Goldman's algorithm which (in linear time) either solves (M) on G, or finds the single block of G which contains all solutions to (M).

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