We consider the problem of computing bounds for causal inference problems with unobserved confounders, where identifiability does not hold. Existing non-parametric approaches for computing such bounds use linear programming (LP) formulations that quickly become intractable for existing solvers because the size of the LP grows exponentially in the number of edges in the underlying causal graph. We show that this LP can be significantly pruned by carefully considering the structure of the causal query, allowing us to compute bounds for significantly larger causal inference problems as compared to what is possible using existing techniques. This pruning procedure also allows us to compute the bounds inclosed form for a special class of causal graphs and queries, which includes a well-studied family of problems where multiple confounded treatments influence an outcome. We also propose a very efficient greedy heuristic that produces very high quality bounds, and scales to problems that are several orders of magnitude larger than those for which the pruned LP can be solved.