The design of approximation algorithms for clustering points in metric spaces is a flourishing area of research, with much effort spent on getting a better understanding of the approximation guarantees possible for many objective functions such as k-median, k-means, and min-sum clustering. This quest for better approximation algorithms is fueled in part by the implicit hope that better approximations will also yield more accurate clusterings. In particular, for problems such as clustering proteins by function, or clustering images by subject, the true aim is to match some unknown correct "target" clustering and the implicit hope is that approximately optimizing these objective functions will in fact produce a clustering that is close to this target in terms of how the data points are clustered. In this paper, we show that if we make this implicit assumption explicit-that is, if we assume our instance satisfies the condition that any c-approximation to the given clustering objective Φ is ǫ-close to the target-then we can efficiently produce clusterings that are O(ǫ)-close to the target, even for values c for which obtaining a c-approximation is NP-hard. We call this condition (c, ǫ)-approximation-stability and show that we can achieve this guarantee for any constant c > 1, for three important and widely studied objective functions, namely k-median, k-means, and min-sum objectives. Thus, we can perform nearly as well in terms of distance to the target clustering as if we could approximate the objective to this NP-hard value. That is, under the assumption that c-approximations are good solutions, we can approximate the target even if in general it is hard to achieve a c-approximation to the objective.
Balcan et al. REF show that in fact this condition can be used to bypass approximation hardness results for a number of clustering objectives including k-median and k-means.
Approximate clustering without the approximation
In SODA ’09: Proceedings of the Nineteenth Annual ACM -SIAM Symposium on Discrete Algorithms
Computer Science