Constant approximation for k-median and k-means with outliers via iterative rounding

Abstract

In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an (α 1 + ≤ 7.081 + )-approximation algorithm for k-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen [16]. For k-means with outliers, we give an (α 2 + ≤ 53.002 + )-approximation, which is the first O(1)-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give α 1 -and (α 1 + )approximation algorithms for matroid and knapsack median problems respectively, improving upon the previous best approximations ratios of 8 [42] and 17.46 [9].The natural LP relaxation for the k-median/k-means with outliers problem has an unbounded integrality gap. In spite of this negative result, our iterative rounding framework shows that we can round an LP solution to an almost-integral solution of small cost, in which we have at most two fractionally open facilities. Thus, the LP integrality gap arises due to the gap between almost-integral and fully-integral solutions. Then, using a pre-processing procedure, we show how to convert an almost-integral solution to a fully-integral solution losing only a constant-factor in the approximation ratio. By further using a sparsification technique, the additive factor loss incurred by the conversion can be reduced to any > 0. * Microsoft Research India. for general and Euclidean metrics respectively. Both problems admit PTASs [2, 21, 19] on fixed-dimensional Euclidean metrics.Despite their simplicity and elegance, a significant shortcoming these formulations face in real-world data sets is that they are not robust to noisy points, i.e., a few outliers can completely change the cost as well as structure of solutions. To overcome this shortcoming, Charikar et al. [12] introduced the robust k-median (RkMed) problem (also called k-median with outliers), which we now define.Definition 1.1 (The Robust k-Median and k-Means Problems). The input to the Robust k-Median (RkMed) problem is a set C of clients, a set F of facility locations, a metric space (C ∪ F, d), integers k and m. The objective is to choose a subset S ⊆ F of cardinality at most k, and a subset C * ⊆ C of cardinality at least m such that the total cost j∈C * d(j, S) is minimized. In the Robust k-Means (RkMeans) problem, we have the same input and the goal is to minimize j∈C * d 2 (j, S).The problem is not just interesting from the clustering point of view. In fact, such a joint view of clustering and removing outliers has been observed to be more effective [15,39] for even the sole task of outlier detection, a very important problem in the real world. Due to these use cases, there has been much recent work [15,23,40] in the applied community on these problems. However, their inherent complexity from the theoretical side is much less understood. For RkMed, Charikar et al. [12] give an algorithm that violates the number of outliers by a factor of (1 + ), and has cost at most 4(1 + 1/ ) times the optimal cost. Chen [1...

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