We study the notion of stability and perturbation resilience introduced by Bilu and Linial (2010) and Awasthi, Blum, and Sheffet (2012). A combinatorial optimization problem is α-stable or α-perturbation-resilient if the optimal solution does not change when we perturb all parameters of the problem by a factor of at most α. We present improved algorithms for stable instances of various clustering and optimization problems, including k-means, k-median, Max Cut, and Minimum Multiway Cut. We also show several hardness results.

The talk is based on joint papers with Haris Angelidakis, Yury Makarychev, and Aravindan Vijayaraghavan.