A New Algorithm for the Single Source Weber Problem with Limited Distances

The single source Weber problem with limited distances (SSWPLD) is a continuous optimization problem in location theory. The SSWPLD algorithms proposed so far are based on the enumeration of all regions of ${\Re }^2$ defined by a given set of n intersecting circumferences. Early algorithms require $O(n^3)$ time for the enumeration, but they were recently shown to be incorrect in case of degenerate intersections, that is, when three or more circumferences pass through the same intersection point. This problem was fixed by a modified enumeration algorithm with complexity $O(n^4)$, based on the construction of neighborhoods of degenerate intersection points. In this paper, it is shown that the complexity for correctly dealing with degenerate intersections can be reduced to $O(n^2\log n)$ so that existing enumeration algorithms can be fixed without increasing their $O(n^3)$ time complexity, which is due to some preliminary computations unrelated to intersection degeneracy. Furthermore, a new algorithm for enumerating all regions to solve the SSWPLD is described: its worst-case time complexity is $O(n^2\log n)$. The new algorithm also guarantees that the regions are enumerated only once.