In this paper, we consider the $k$-center/median/means clustering with outliers problems (or the $(k, z)$-center/median/means problems) in the distributed setting. Most previous distributed algorithms have their communication costs linearly depending on $z$, the number of outliers. Recently Guha et al.[10] overcame this dependence issue by considering bi-criteria approximation algorithms that output solutions with $2z$ outliers. For the case where $z$ is large, the extra $z$ outliers discarded by the algorithms might be too large, considering that the data gathering process might be costly. In this paper, we improve the number of outliers to the best possible $(1+\epsilon)z$, while maintaining the $O(1)$-approximation ratio and independence of communication cost on $z$. The problems we consider include the $(k, z)$-center problem, and $(k, z)$-median/means problems in Euclidean metrics. Implementation of the our algorithm for $(k, z)$-center shows that it outperforms many previous algorithms, both in terms of the communication cost and quality of the output solution.

The paper describes algorithms for distributed k-center/k-median/k-means clustering. For distributed k-center, it is known that any constant approximation needs Omega(z) communication cost, where z is the number of allowed outliers. This can be avoided by using a bicriteria approximation where more than z outliers are allowed, but the cost is compared to the best solution with at most z outliers. It is known that using 2z outliers is sufficient to obtain a O(1)-approximation. The paper at hand now improves the violation with respect to the number of outliers to (1 eps)*z.

In this paper, we consider the $k$-center/median/means clustering with outliers problems (or the $(k, z)$-center/median/means problems) in the distributed setting. Most previous distributed algorithms have their communication costs linearly depending on $z$, the number of outliers. Recently Guha et al.[10] overcame this dependence issue by considering bi-criteria approximation algorithms that output solutions with $2z$ outliers. For the case where $z$ is large, the extra $z$ outliers discarded by the algorithms might be too large, considering that the data gathering process might be costly. In this paper, we improve the number of outliers to the best possible $(1 \epsilon)z$, while maintaining the $O(1)$-approximation ratio and independence of communication cost on $z$.