## Abstract

We consider the following clustering problem: we have a complete graph on *n* vertices (items), where each edge (*u*, *v*) is labeled either + or − depending on whether *u* and *v* have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as much as possible with the edge labels. That is, we want a clustering that maximizes the number of + edges within clusters, plus the number of − edges between clusters (equivalently, minimizes the number of disagreements: the number of − edges inside clusters plus the number of + edges between clusters). 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.

An interesting feature of this clustering formulation is that one does not need to specify the number of clusters *k* as a separate parameter, as in measures such as *k*-median or min-sum or min-max clustering. Instead, in our formulation, the optimal number of clusters could be any value between 1 and *n*, depending on the edge labels. We look at approximation algorithms for both minimizing disagreements and for maximizing agreements. For minimizing disagreements, we give a constant factor approximation. For maximizing agreements we give a PTAS, building on ideas of Goldreich, Goldwasser, and Ron (1998) and de la Veg (1996). We also show how to extend some of these results to graphs with edge labels in [−1, +1], and give some results for the case of random noise.

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