Local Guarantees in Graph Cuts and Clustering

  title={Local Guarantees in Graph Cuts and Clustering},
  author={Moses Charikar and Neha Gupta and Roy Schwartz},
Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min \(\,s-t\,\) Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled \(+\) or − and the goal is to produce a clustering that agrees with the labels as much as possible: \(+\) edges within clusters and − edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize… 

Ju n 20 19 Min-Max Correlation Clustering via MultiCut ⋆

The first nontrivial approximation algorithm for correlation clustering is provided, achieving an O( √ logn ·max{log(|E|), log(k)}) approximation for general weighted graphs, where |E−| denotes the number of negative edges and k is thenumber of clusters in the optimum solution.

Correlation clustering with local objectives

This paper presents the first known algorithm for minimizing the \ell_q norm of the disagreements vector on arbitrary graphs and provides an improved algorithm for minimize the\ell-q norm (q >= 1) of the disagreement vector on complete graphs.

Improved algorithms for Correlation Clustering with local objectives

The first known algorithm for minimizing the $\ell_q$ norm of the disagreements vector on arbitrary graphs is presented and an improved algorithm for minimize the $q \geq 1$ norm on complete graphs is provided.

Correlation Clustering with Sherali-Adams

This paper shows that there exists a (1 . 994+ ε )-approximation algorithm based on O (1 /ε 2 ) rounds of the Sherali-Adams hierarchy and reaches an approximation ratio of 2 + ε for Correlation Clustering.

𝓁p-norm Multiway Cut

It is shown that `p-norm-multiway-cut is NP-hard for constant number of terminals and isNP-hard in planar graphs and on the algorithmic side, an O(log n)approximation for all p ≥ 1 is designed.

lp-Norm Multiway Cut

It is shown that lp-norm-multiway-cut is NP-hard for constant number of terminals and isNP-hard in planar graphs and an O(log2 n)-approximation for all p ≥ 1 is designed.

Min-Max Correlation Clustering via MultiCut

The goal is to obtain a partitioning of the vertices that minimizes disagreements – weight of negative edges trapped inside a cluster plus positive edges between different clusters.

Fixed Parameter Approximation Scheme for Min-max k-cut

The study of Minmax $k-cut is initiated by showing that it is NP-hard and W[1]-hard when parameterized by $k, and a parameterized approximation scheme when parameterization by $ k is designed.

(cid:2) p -Norm Multiway Cut

(cid:2) p - norm - multiway - cut is shown to be NP-hard forconstantnumberofterminals and is NP- hard in planar graphs.

Local Correlation Clustering with Asymmetric Classification Errors

An O ( (1/α)/2−/2p · log 1/α ) approximation algorithm is given for Correlation Clustering and an almost matching convex programming integrality gap is shown.



Correlation Clustering

This formulation is motivated from a document clustering problem in which one has a pairwise similarity function f learned from past data, and the goal is to partition the current set of documents in a way that correlates with f as much as possible; it can also be viewed as a kind of “agnostic learning” problem.

Correlation Clustering: maximizing agreements via semidefinite programming

This work gives a 0.7666-approximation algorithm for maximizing agreements on any graph even when the edges have non-negative weights (along with labels) and they want to maximize the weight of agreements.

Min-max Graph Partitioning and Small Set Expansion

An O(√log n log (1/p) bicriteria approximation algorithm for the general case of Small Set Expansion and O(1) approximation algorithms for graphs that exclude any fixed minor.

Correlation clustering in general weighted graphs

Improved Approximation Algorithms for Bipartite Correlation Clustering

The first algorithm is a linear program based $4-approximation algorithm, which requires solving a large convex problem, which becomes prohibitive even for modestly sized tasks.

Clustering with qualitative information

This work considers the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity, and gives a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs.

Correlation Clustering and Biclustering With Locally Bounded Errors

This work provides a rounding algorithm which converts “fractional clusterings” into discrete clusterings while causing only a constant-factor blowup in the number of errors at each vertex.

Approximate max-flow min-(multi)cut theorems and their applications

The proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of Leighton and Rao and Klein et al. and thereby obtain an improved bound for the latter problem.

Near Optimal LP Rounding Algorithm for CorrelationClustering on Complete and Complete k-partite Graphs

These results improve a long line of work on approximation algorithms for correlation clustering in complete graphs, previously culminating in a ratio of 2.5 by Ailon, Charikar and Newman.

Simple Local Search Problems That are Hard to Solve

It is shown here that several natural, simple local search problems are PLS-complete, and thus just as hard.