Georgia Institute of Technology College of Sciences

Recently, Bilu and Linial formalized an implicit assumption often made when choosing a clustering objective: that the optimum clustering to the objective should be preserved under small multiplicative perturbations to distances between points. They showed that for max-cut clustering it is possible to circumvent NP-hardness and obtain polynomial-time algorithms for instances resilient to large (factor O(\sqrt{n})) perturbations, and subsequently Awasthi et al. considered center-based objectives, giving algorithms for instances resilient to O(1) factor perturbations. In this talk, for center-based objectives, we present an algorithm that can optimally cluster instances resilient to (1+\sqrt{2})-factor perturbations, solving an open problem of Awasthi et al. For k-median, a center-based objective of special interest, we additionally give algorithms for a more relaxed assumption in which we allow the optimal solution to change in a small fraction of the points after perturbation. We give the first bounds known for k-median under this more realistic and more general assumption. We also provide positive results for min-sum clustering which is a generally much harder objective than center-based objectives. Our algorithms are based on new linkage criteria that may be of independent interest.