Similar Elements and Metric Labeling on Complete Graphs

We consider a problem that involves finding similar elements in a collection of sets. The problem is motivated by applications in machine learning and pattern recognition. We formulate the similar elements problem as an optimization and give an efficient approximation algorithm that finds a solution within a factor of 2 of the optimal. The similar elements problem is a special case of the metric labeling problem and we also give an efficient 2-approximation algorithm for the metric labeling problem on complete graphs.