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The perfect phylogeny problem is of central importance to both evolutionary biology and population genetics. Missing values are a common occurrence in both sequence and genotype data, but they make the problem of finding a perfect phylogeny NPhard even for binary characters. We introduce new and efficient perfect phylogeny algorithms for broad classes of binary and multistate data with missing values. Specifically, we address binary missing data consistent with the rich data hypothesis (RDH) introduced by Halperin and Karp and give an efficient algorithm for enumerating phylogenies. This algorithm is useful for computing the probability of data with missing values under the coalescent model. In addition, we use the partition intersection (PI) graph and chordal graph theory to generalize the RDH to multi-state characters with missing values. For a bounded number of states, we provide a fixed parameter tractable algorithm for the perfect phylogeny problem with missing data. Utilizing the PI graph, we are able to show that under multiple biologically motivated models for character data, our generalized RDH holds with high probability, and we evaluate our results with extensive empirical analysis.