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Approximation Algorithms for Clustering Problems Open Access


Other title
minimum sum of radii
facility location
minimum sum of diameters
approximation algorithms
Type of item
Degree grantor
University of Alberta
Author or creator
Behsaz, Babak
Supervisor and department
Salavatipour, Mohammad R. (Computing Science)
Examining committee member and department
Koenemann, Jochen (Combinatorics and Optimization, University of Waterloo)
Hayward, Ryan B. (Computing Science)
Ardakani, Masoud (Electrical and Computer Engineering)
Stewart, Lorna (Computing Science)
Department of Computing Science

Date accepted
Graduation date
Doctor of Philosophy
Degree level
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. 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 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 d_j, each facility i has capacity u and opening cost f_i, and a metric cost c_{ij} 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 (\alpha,\beta)-approximation algorithms, where these algorithms return a solution whose cost is within factor \alpha of the optimum and violates the capacity constraints within factor \beta. We present the first constant approximations with violation factor less than 2. In addition, we present a quasi-polynomial time (1+\epsilon,1+\epsilon)-approximation for the (uniform) UCFLP in Euclidean metrics, for any constant \epsilon > 0.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
B. Behsaz, Mohammad R. Salavatipour, and Z. Svitkina, "New Approximation Algorithms for the Unsplittable Capacitated Facility Location Problem," in proceedings of 13th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT’12), Pages 237-248, 2012.B. Behsaz and Mohammad R. Salavatipour, "On Minimum Sum of Radii and Diameters Clustering," in proceedings of 13th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT’12), Pages 71-82, 2012.

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