# 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} }

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

### Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations

- Computer ScienceAPPROX-RANDOM
- 2018

This model shows that the unified MST and dynamic programming based algorithm proposed by Angelidakis et.

### k-center Clustering under Perturbation Resilience

- Computer ScienceICALP
- 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.

### Clustering Perturbation Resilient Instances

- Computer ScienceArXiv
- 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.

### Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means

- Computer Science, MathematicsSODA
- 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.

### Certified Algorithms: Worst-Case Analysis and Beyond

- Computer ScienceITCS
- 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.

### On the Local Structure of Stable Clustering Instances

- Computer Science2017 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.

### On Perturbation Resilience of Non-uniform k-Center

- MathematicsAPPROX-RANDOM
- 2020

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.

### Bilu-Linial stability, certified algorithms and the Independent Set problem

- Computer Science, MathematicsESA
- 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.

### On Euclidean k-Means Clustering with α-Center Proximity

- 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.

### Stability and Recovery for Independence Systems

- Computer ScienceESA
- 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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