An Optimal Algorithm for an Outerplanar Facility Location Problem with Improved Time Complexity

We consider a network facility location problem with unbounded production levels. This problem is NP-hard in the general case and is known to have an optimal solution with quadratic complexity on a tree network. We study the case of a network representable by an outerplanar graph, i.e., by a graph whose vertices belong to one (outer) face. This problem is known to have an optimal algorithm with time complexity O(nm3), where n is the number of vertices and m is the number of possible facility locations. Using some properties of outerplanar graphs (binary 2-trees) and the existence of an optimal solution with a family of centrally connected service areas, we obtain recurrence relations for the construction of an optimal algorithm with time complexity that is smaller by a factor of m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt m $$\end{document} than the time complexity of the earlier algorithm.