  We show that there is a largely unexplored class of functions (positive
polymatroids) that can define proper discrete metrics over pairs of binary
vectors and that are fairly tractable to optimize over. By exploiting
submodularity, we are able to give hardness results and approximation
algorithms for optimizing over such metrics. Additionally, we demonstrate
empirically the effectiveness of these metrics and associated algorithms on
both a metric minimization task (a form of clustering) and also a metric
maximization task (generating diverse k-best lists).
