# Algorithms for stable and perturbation-resilient problems

@article{Angelidakis2017AlgorithmsFS,
title={Algorithms for stable and perturbation-resilient problems},
author={Haris Angelidakis and Konstantin Makarychev and Yury Makarychev},
journal={Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing},
year={2017}
}
• Published 19 June 2017
• Computer Science, Mathematics
• Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
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. In this paper, we give improved algorithms for stable instances of various clustering and combinatorial optimization problems. We also prove several hardness results. We…
43 Citations
• Computer Science
APPROX-RANDOM
• 2018
This model shows that the unified MST and dynamic programming based algorithm proposed by Angelidakis et.
• Computer Science
ICALP
• 2016
This work provides strong positive results both for the asymmetric and symmetric k-center problems under a natural input stability (promise) condition called α-perturbation resilience and provides algorithms that give strong guarantees simultaneously for stable and non-stable instances.
• Computer Science
ArXiv
• 2018
This work considers stable instances of Euclidean $k-means that have provable polynomial time algorithms for recovering optimal cluster and proposes simple algorithms with running time linear in the number of points and the dimension that provably recover the optimal clustering. • Computer Science, Mathematics SODA • 2019 It is shown that for any fixed \epsilon>0, (1+\ep silon)-stable instances of k-Means in doubling metrics can be solved in polynomial time and under a plausible PCP hypothesis this is essentially tight. • Computer Science ITCS • 2020 The notion of a certified algorithm is introduced, its properties are described, a framework for designing certified algorithms is presented, and examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means are provided. • Computer Science 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) • 2017 It is obtained that the widely-used Local Search algorithm has strong performance guarantees for both the tasks of recovering the underlying optimal clustering and obtaining a clustering of small cost. The NUkC problem under perturbation resilience, which was introduced by Bilu and Linial, is shown to be polynomial time solvable when the r_i$'s belong to a constant sized set, however, it is shown that perturbations resilience does not help in the general case.
• Computer Science, Mathematics
ESA
• 2019
A general result is proved showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances of Node Multiway Cut.
• Computer Science
• 2019
It is shown that for any α′ > 1, there exists an α 6 α′, (α > 1), and an ε0 > 0 such that minimizing the k-means objective over clusterings that satisfy α-center proximity is NP-hard to approximate within a multiplicative (1 +ε0) factor.
• Computer Science
ESA
• 2017
This work considers perturbation-stable instances, in the sense of Bilu and Linial, and precisely identifies the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution, and resolves the worst-case approximation guarantee of local search in p-extendible systems.

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