The k -Centrum Straight-line Location Problem

A line is sought in the plane which minimizes the sum of the k largest (Euclidean) weighted distances from n given points. This problem generalizes the known straight-line center and median problems and, as far as the authors are aware, has not been tackled up to now. By way of geometric duality it is shown that such a line may always be found which either passes through two of the given points or lying at equal weighted distance from three of these. This allows construction of an algorithm to find all t -centrum lines for 1 ≤ t ≤ k in O (( k + log n ) n 3 ). Finally it is shown that both, the characterization of an optimal line and the algorithm, can be extended to any smooth norm.