Are Stable Instances Easy?

  title={Are Stable Instances Easy?},
  author={Yonatan Bilu and Nathan Linial},
  journal={Combinatorics, Probability and Computing},
  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. 

On the practically interesting instances of MAXCUT

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.

Perturbation resilience for the facility location problem

Stability and Recovery for Independence Systems

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

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

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

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

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

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

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

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.



Spectral partitioning of random graphs

  • Frank McSherry
  • Computer Science, Mathematics
    Proceedings 2001 IEEE International Conference on Cluster Computing
  • 2001
This paper shows that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density.

Clusterability: A Theoretical Study

This work addresses measures of the clusterability of data sets with generality, aiming for conclusions that apply regardless of any particular clustering algorithm or any specic data generation model, as well as proposing a new notion of data clusterability.

Graph bisection algorithms with good average case behavior

A polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output is described, which shows that the algorithm chooses the former course with high probability for many natural classes of graphs.

The ellipsoid method and its consequences in combinatorial optimization

The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.

The Effectiveness of Lloyd-Type Methods for the k-Means Problem

This work investigates variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and proposes and justifies a clusterability criterion for data sets.

Heuristics for Semirandom Graph Problems

It is shown that when p<(1??)lnn /?n, an independent set of size |S| cannot be recovered, unless NP?BPP, and a heuristic is given that recovers this bisection with high probability when p?q?cplogn/n, for c a sufficiently large constant.

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

Algorithms for graph partitioning on the planted partition model

The NP-hard graph bisection problem is to partition the nodes of an undirected graph into two equal-sized groups so as to minimize the number of edges that cross the partition. The more general graph

The Metropolis Algorithm for Graph Bisection

Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time

The smoothed analysis of algorithms is introduced, which is a hybrid of the worst-case and average-case analysis of algorithm performance and shows that the shadow-vertex simplex algorithm has polynomial smoothed complexity.