  We propose novel methods for max-cost Discrete Function Evaluation Problem
(DFEP) under budget constraints. We are motivated by applications such as
clinical diagnosis where a patient is subjected to a sequence of (possibly
expensive) tests before a decision is made. Our goal is to develop strategies
for minimizing max-costs. The problem is known to be NP hard and greedy methods
based on specialized impurity functions have been proposed. We develop a broad
class of \emph{admissible} impurity functions that admit monomials, classes of
polynomials, and hinge-loss functions that allow for flexible impurity design
with provably optimal approximation bounds. This flexibility is important for
datasets when max-cost can be overly sensitive to "outliers." Outliers bias
max-cost to a few examples that require a large number of tests for
classification. We design admissible functions that allow for accuracy-cost
trade-off and result in $O(\log n)$ guarantees of the optimal cost among trees
with corresponding classification accuracy levels.
