Reconstructing Mutational History in Multiply Sampled Tumors Using Perfect Phylogeny Mixtures

High-throughput sequencing of cancer genomes have motivated the problem of inferring the ancestral history of somatic mutations that accumulate in cells during cancer progression. While the somatic mutation process in cancer cells meets the requirements of the classic Perfect Phylogeny problem, nearly all cancer sequencing studies do not sequence single cancerous cells, but rather thousands-millions of cells in a tumor sample. In this paper, we formulate the Perfect Phylogeny Mixture problem of inferring a perfect phylogeny given somatic mutation data from multiple tumor samples, each of which is a superposition of cells, or “species.” We prove that the Perfect Phylogeny Mixture problem is NP-hard, using a reduction from the graph coloring problem. Finally, we derive an algorithm to solve the problem.