Correlation Clustering Revisited: The "True" Cost of Error Minimization Problems

Correlation Clustering was defined by Bansal, Blum, and Chawla is the problem of clustering a set of elements based on a, possibly inconsistent., binary similarity function between element; pairs. Their setting is agnostic in the sense that a ground truth clustering is not assumed to exist, and the cost of a solution is computed against the input similarity function. This problem has been studied in theory and in practice and has been subsequently proven to be APX-Hard. In this work we assume that there does exist all unknown correct clustering of the data. In this setting, we argue that it is more reasonable to measure the output clustering's accuracy against the unknown underlying true clustering. We present two main results. The first is a novel method for continuously morphing a general (non-metric) function into a pseudometric. This technique may be useful for other metric embedding and clustering problems. The second is a simple algorithm for randomly rounding a pseudometric into a clustering. Combining the two, we obtain a certificate for the possibility of getting a solution of factor strictly less than 2 for our problem. This approximation coefficient; could not have been achieved by considering the agnostic version of the problem unless P = NP.