Local Guarantees in Graph Cuts and Clustering

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 -approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the -approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games.

M. Charikar and N. Gupta—Supported by NSF grants CCF-1617577, CCF-1302518 and a Simons Investigator Award.

R. Schwartz—Supported by ISF grant 1336/16.