  We provide algorithms for isotonic regression minimizing $L_0$ error (Hamming
distance). This is also known as monotonic relabeling, and is applicable when
labels have a linear ordering but not necessarily a metric. There may be
exponentially many optimal relabelings, so we look at secondary criteria to
determine which are best. For arbitrary ordinal labels the criterion is
maximizing the number of labels which are only changed to an adjacent label
(and recursively apply this). For real-valued labels we minimize the $L_p$
error. For linearly ordered sets we also give algorithms which minimize the sum
of the $L_p$ and weighted $L_0$ errors, a form of penalized (regularized)
regression. We also examine $L_0$ isotonic regression on multidimensional
coordinate-wise orderings. Previous algorithms took $\Theta(n^3)$ time, but we
reduce this to $o(n^{3/2})$.
