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

Correlation Clustering was defined by Bansal, Blum, and Chawla as 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 only reasonable way to measure the cost of a solution is by comparing it with the input similarity function. This problem has been studied in theory and application and has been subsequently proven to be APX-Hard. In this work we assume that there does exist an unknown correct clustering of the data. This is the case in applications such as record linkage in databases. In this setting, we argue that it is more reasonable to measure accuracy of the output clustering against the unknown underlying true clustering. This corresponds to the intuition that in real life an action is penalized or rewarded based on reality and not on our noisy perception thereof. The traditional combinatorial optimization version of the problem only offers an indirect solution to our revisited version via a triangle inequality argument applied to the distances between the output clustering, the input similarity function and the underlying ground truth. In the revisited version, we show that it is possible to shortcut the traditional optimization detour and obtain a factor 2 approximation. This factor could not have possibly been obtained by using a solution to the traditional problem as a black box, unless it was an exact optimal solution. Our result therefore shortcuts the APX-Hardness, and could be useful for revisiting many other combinatorial optimization problems. Our analysis consists of two solutions. The first gives a simple 2-approximation algorithm. The second involves a novel way to continuously morph a general (non-metric) distance function into a metric. This technique is interesting in its own right and may be useful for other metric embedding problems. The resulting morphed solution is randomly rounded into a clustering. En route, in certain cases we obtain a certificate for the possibility of getting a solution of factor strictly less than 2. Finally, we show simple cases in which randomness is necessary for achieving a solution of factor strictly less than 2, thus justifying the use of randomization in our solution.