GIS software has some pretty sophisticated geoprocessing algorithms to do this sort of thing, and although numerical and iterative and recursive in nature (as opposed to analytical) they do a very good job. I used to write this kind of software years ago, but I must confess the stuff that's available now is so good I wouldn't even know where to begin today. It's magic.

The limiting factor is not the algorithm, it's the data. The best spatial population data we have is from Census tracts, and these polygons are widely heterogeneous in shape and size (they're designed to guide census takers). If a voting district boundary crosses a census tract boundary you're going to have to do a space-based extrapolation (actually, an area-weighted average in 2-D) to give a fair estimate of the local population distribution. The software can do this but the politicians will always point to it and say it's not fair. It's not the first time valid statistical extrapolations of census data have been rejected in order to preserve the status quo.