The work here studies the communication cost for a multi-server multi-task distributed computation framework, and does so for a broad class of functions and data statistics. Considering the framework where a user seeks the computation of multiple complex (conceivably non-linear) tasks from a set of distributed servers, we establish communication cost upper bounds for a variety of data statistics, function classes and data placements across the servers. To do so, we proceed to apply, for the first time here, Körner's characteristic graph approach -- which is known to capture the structural properties of data and functions -- to the promising framework of multi-server multi-task distributed computing. Going beyond the general expressions, and in order to offer clearer insight, we also consider the well-known scenario of cyclic dataset placement and linearly separable functions over the binary field, in which case our approach exhibits considerable gains over the state of art. Similar gains are identified for the case of multi-linear functions.