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Local Guarantees in Graph Cuts and Clustering

  • Moses Charikar
  • Neha Gupta
  • Roy Schwartz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)

Abstract

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 a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max \(s-t\) Cut: find an \(s-t\) cut minimizing the largest number of cut edges incident on any node. We present the following results: (1) an \(O(\sqrt{n})\)-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems), (2) a remarkably simple 7-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of 48), and (3) a Open image in new window -approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the Open image in new window -approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games.

Keywords

Approximation algorithms Graph cuts Correlation clustering Linear programming 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.TechnionHaifaIsrael

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