Weighted isotonic regression under the L1 norm

Isotonic regression, the problem of finding values that best fit given observations and conform to specific ordering constraints, has found many applications in biomedical research and other fields. When the constraints form a partial ordering, solving the problem under the L1 error measure takes O(n) when there are n observations. The analysis of large-scale microarray data, which is one of the important tools in biology, using isotonic regression is hence expensive. This is because in microarray analysis, the same procedure is used for studying the fit of tens of thousands of genes to a given partial order. Fast estimation for the fitting error is therefore highly desired to reduce the number of regression instances through pruning. In this paper, we present approximation algorithms to the isotonic regression problem under the L1 error measure. We relate the problem to an edge packing problem and in the special case when the observations are not weighted, we relate it to a weighted