Approximation algorithms for clustering and facility location problems

In this thesis we design and analyze algorithms for various facility location and clustering problems. The problems we study are NP-Hard and therefore, assuming P is not equal NP, there do not exist polynomial time algorithms to solve them optimally. One approach to cope with the intractability of these problems is to design approximation algorithms which run in polynomial-time and output a near-optimal solution for all instances of the problem. However these algorithms do not always work well in practice. Often heuristics with no explicit approximation guarantee perform quite well. To bridge this gap between theory and practice, and to design algorithms that are tuned for instances arising in practice, there is an increasing emphasis on beyond worst-case analysis. In this thesis we consider both these approaches. In the first part we design worst case approximation algorithms for Uniform Submodular Facility Location (USFL), and Capacitated k-center (CapKCenter) problems. USFL is a generalization of the well-known Uncapacitated Facility Location problem. In USFL the cost of opening a facility is a submodular function of the clients assigned to it (the function is identical for all facilities). We show that a natural greedy algorithm (which gives constant factor approximation for Uncapacitated Facility Location and other facility location problems) has a lower bound of log(n), where n is the number of clients. We present an O(log^2 k) approximation algorithm where k is the number of facilities. The algorithm is based on rounding a convex relaxation. We further consider several special cases of the problem and give improved approximation bounds for them. The CapKCenter problem is an extension of the well-known k-center problem: each facility has a maximum capacity on the number of clients that can be assigned to it. We obtain a 9-approximation for this problem via a linear programming (LP) rounding procedure. Our result, combined with previously known lower bounds, almost settles the integrality gap for a natural LP relaxation. In the second part we consider several well-known clustering problems like k-center, k-median, k-means and their corresponding outlier variants. We use beyond worst-case analysis due to the practical relevance of these problems. In particular we show that when the input instances are 2-perturbation resilient (i.e. the optimal solution does not change when the distances change by a multiplicative factor of 2), the LP integrality gap for k-center (and also asymmetric k-center) is 1. We further introduce a model of perturbation resilience for clustering with outliers. Under this new model, we show that previous results (including our LP integrality result) known for clustering under perturbation resilience also extend for clustering with outliers. This leads to a dynamic programming based heuristic for k-means with outliers (k-means-outlier) which gives an optimal solution when the instance is 2-perturbation resilient. We propose two more algorithms for k-means-outlier — a sampling based algorithm which gives an O(1) approximation when the optimal clusters are not “too small”, and an LP rounding algorithm which gives an O(1) approximation at the expense of violating the number of clusters and outliers by a small constant. We empirically study our proposed algorithms on several clustering datasets