- Michael R. Fellows
- Tzvika Hartman
- Danny Hermelin
- Gad M. Landau
- Frances Rosamond
- Liat Rozenberg

## Abstract

The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set of genotypes. This problem is of enormous importance in many practical fields, such as the investigation of diseases, or other types of genetic mutations. In order to find the haplotypes that are as close as possible to the real set of haplotypes that comprise the genotypes, two models have been suggested which by now have become widely accepted: The perfect phylogeny model and the pure parsimony model. All known algorithms up till now for the above problem may find haplotypes that are not necessarily plausible, i.e. very rare haplotypes or haplotypes that were never observed in the population. In order to overcome this disadvantage we study in this paper, for the first time, a new constrained version of HIP under the above mentioned models. In this new version, a pool of plausible haplotypes ~H is given together with the set of genotypes G, and the goal is to find a subset $H \subseteq \widetilde{H}$ that resolves G. For the constrained perfect phylogeny haplotyping (CPPH) problem we provide initial insights and polynomial-time algorithms for some restricted cases that help understanding the complexity of that problem. We also prove that the constrained parsimony haplotyping (CPH) problem is fixed parameter tractable by providing a parameterized algorithm that applies an interesting dynamic programming technique for solving the problem.

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