Min-Max Correlation Clustering via MultiCut

Abstract

Correlation clustering is a fundamental combinatorial optimization problem arising in many contexts and applications that has been the subject of dozens of papers in the literature. In this problem we are given a general weighted graph where each edge is labeled positive or negative. The goal is to obtain a partitioning (clustering) of the vertices that minimizes disagreements -weight of negative edges trapped inside a cluster plus positive edges between different clusters. Most of the papers on this topic mainly focus on minimizing total disagreement, a global objective for this problem. In this paper we study a cluster-wise objective function that asks to minimize the maximum number of disagreements of each cluster, which we call min-max correlation clustering. The min-max objective is a natural objective that respects the quality of every cluster. In this paper, we provide the first nontrivial approximation algorithm for this problem achieving an O( log n · max{log(|E − |), log(k)}) approximation for general weighted graphs, where |E − | denotes the number of negative edges and k is the number of clusters in the optimum solution. To do so, we also obtain a corresponding result for multicut where we wish to find a multicut solution while trying to minimize the total weight of cut edges on every component. The results are then further improved to obtain (i) O(r 2 )-approximation for min-max correlation clustering and min-max multicut for graphs that exclude Kr,r minors (ii) a 14-approximation for the min-max correlation clustering on complete graphs.

Local Correlation Clustering with Asymmetric Classification Errors

“…We use the standard definition of the ℓ p norm of a vector x: x p = ( u |x u | p ) 1 p . Since its introduction by Puleo and Milenkovic (2018), local objectives for Correlation Clustering have been mainly studied under two models (see Charikar, Gupta, and Schwartz (2017), Ahmadi, Khuller, and Saha (2019), Kalhan, Makarychev, and Zhou (2019)). We will refer to these models as (1) Correlation Clustering on Complete Graphs, and (2) Correlation Clustering with Noisy Partial Information.…”

Improved algorithms for Correlation Clustering with local objectives

“…Recently, Ahmadi, Khuller, and Saha [2019] introduced an alternative min max objective for correlation clustering (which we call AKS min max objective). For a cluster C ⊆ V , let us refer to similar edges with exactly one endpoint in C and dissimilar edges with both endpoints in C as edges in disagreements with respect to C. We call the weight of all edges in disagreement with C the cost of C. Then, the AKS min max objective asks to find a clustering C 1 , .…”

“…, C T that minimizes the maximum cost C i . Ahmadi et al [2019] give an O(log n) approximation algorithm for this objective.…”

“…In this paper, we also consider the AKS min max objective. For this objective, we give a (2 + ε) approximation algorithm, which improves the approximation ratio of O(log n) given by Ahmadi, Khuller, and Saha [2019].…”

“…The presented a 10 approximation algorithm for this problem, which was improved to 7 by Charikar, Gupta, and Schwartz [2017]. Recently, Ahmadi et al [2019] studied an alternative objective for the correlation clustering problem. Motivated by creating balanced communitites for problems such as image segmentation and community detection in social networks, they propose a new cluster-wise min-max objective.…”