  Consider a geometric range space $(X,\c{A})$ where each data point $x \in X$
has two or more values (say $r(x)$ and $b(x)$). Also consider a function
$\Phi(A)$ defined on any subset $A \in (X,\c{A})$ on the sum of values in that
range e.g., $r_A = \sum_{x \in A} r(x)$ and $b_A = \sum_{x \in A} b(x)$. The
$\Phi$-maximum range is $A^* = \arg \max_{A \in (X,\c{A})} \Phi(A)$. Our goal
is to find some $\hat{A}$ such that $|\Phi(\hat{A}) - \Phi(A^*)| \leq
\varepsilon.$ We develop algorithms for this problem for range spaces with
bounded VC-dimension, as well as significant improvements for those defined by
balls, halfspaces, and axis-aligned rectangles. This problem has many
applications in many areas including discrepancy evaluation, classification,
and spatial scan statistics.
