Approximation algorithms for clustering problems

Abstract

In this thesis, we present some approximation algorithms for the following clustering problems: Minimum Sum of Radii (MSR), Minimum Sum of Diameters (MSD), and Unsplittable Capacitated Facility Location (UCFL). Given a metric (V,d) and an integer k, we consider the problem of partitioning the points of V into k clusters so as to minimize the sum of radii (MSR) or the sum of diameters (MSD) of these clusters. We call a cluster containing a single point, a singleton cluster. For the MSR problem when singleton clusters are not allowed, we give an exact algorithm for metrics induced by unweighted graphs. For the MSD problem on the plane with Euclidean distances, we present a polynomial time approximation scheme. In addition, we settle the open problem of complexity of the MSD problem with constant k by giving a polynomial time exact algorithm in this case. In the (uniform) UCFL problem, we are given a set of clients and a set of facilities where client j has demand dj, each facility i has capacity u and opening cost fi, and a metric cost cij which denotes the cost of serving one unit of demand of client j at facility i. The goal is to open a subset of facilities and assign each client to exactly one open facility so that the total amount of demand assigned to each open facility is no more than u, while minimizing the total cost of opening facilities and serving clients. As it is NP-hard to give a solution without violating the capacities, we consider bicriteria (α,β)-approximation algorithms, where these algorithms return a solution whose cost is within factor α of the optimum and violates the capacity constraints within factor β. We present the first constant approximations with violation factor less than 2. In addition, we present a quasipolynomial time (1 + e; 1 + e)-approximation for the (uniform) UCFLP in Euclidean metrics, for any constant e > 0.