# Are Stable Instances Easy?

@article{Bilu2009AreSI, title={Are Stable Instances Easy?}, author={Yonatan Bilu and Nathan Linial}, journal={Combinatorics, Probability and Computing}, year={2009}, volume={21}, pages={643 - 660} }

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

## 122 Citations

### On the practically interesting instances of MAXCUT

- Computer ScienceSTACS
- 2013

This work investigates practically interesting instances of MAXCUT, viewed as a clustering problem, and shows how to solve in polynomial time distinguished, metric, expanding and dense instances ofMAXCUT under mild stability assumptions.

### 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.

### Polynomial Time Algorithm for 2-Stable Clustering Instances

- Computer Science, MathematicsArXiv
- 2016

This paper provides a polynomial time algorithm for $2-stable instances, improving on and answering an open question in ~\cite{Balcan12}.

### 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.

### Clustering Stable Instances of Euclidean k-means

- Computer ScienceNIPS
- 2017

This work designs efficient algorithms that provably recover the optimal clustering for instances that are additive perturbation stable and shows an efficient algorithm with provable guarantees that is also robust to outliers.

### Beyond worst-case analysis in approximation algorithms

- Computer Science
- 2012

The results suggest that the approximability of the Densest k-subgraph problem may be similar from both worst-case and average-case perspectives, in contrast to graph partitioning.

### Semi-Supervised Clustering of stable instances

- Computer Science
- 2018

This work designs efficient algorithms which solve problems of multiplicative perturbation stability using a noisy oracle model, and designs an oracle O which answers pairwise queries.

### Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

- Computer Science, MathematicsSODA
- 2014

It is proved that there is no robust polynomial-time algorithm for γ-stable instances of Max Cut when γ < α SC(n/2), where αSC is the best approximation factor for Sparsest Cut with non-uniform demands, and it is shown that the standard SDP relaxation for Max Cut is integral if [EQUATION].

### On the Geometry of Stable Steiner Tree Instances

- MathematicsCCCG
- 2022

This note gives strong geometric structural properties that need to be satisfied by stable instances of Steiner trees and makes use of, and strengthen, these geometric properties to show that 1.562-stable instances of Euclidean Steiner Trees are polynomialtime solvable.

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