Given discrete rvs X,Y, find Z s.t. I(Z;X) is high and I(Z;Y) is low. -- known problem? Consider the following problem. Let $X$ and $Y$ be discrete random variables. The goal is to find a random variable $Z$ such that, informally, $I(Z;X)$ is high and $I(Z;Y)$ is low.

More precisely, either: $~\max_Z I(Z;X) ~\text{s.t.}~ I(Z;Y) \le c~$ or $~\min_Z I(Z;Y) ~\text{s.t.}~ I(Z;X) \ge c~$ (for some constant $c$).

This seems like a very natural, fundamental (if toy?) problem, so surely something is known about it. (I know that for $X,Y$ both with finite support, (large) linear programs can be written to solve for such $Z$, but I imagine there's a neater way.) Any pointers would be appreciated.

Thanks!
In one form, your question is actually a clustering question that is addressed by the information bottleneck method. 

Roughly, $Y$ represents the rows of a joint distribution and $X$ represents the columns. The random variable $Z$ then represents a distribution over the rows ($Y$) that is highly compressed (because $I(Z;Y)$ is small) but represents the joint distribution well (because $I(Z;X)$ is large). 

It's not clear to me that this can be solved using LPs. the underlying problem is not convex. But there's a simple EM-style alternating optimization scheme that can be used to solve it.