Tractable cases of $(*,2)$-bounded parsimony haplotyping

Parsimony haplotyping is the problem of finding a set of haplotypes of minimum cardinality that explains a given set of genotypes, where a genotype is explained by two haplotypes if it can be obtained as a combination of the two. This problem is NP-complete in the general case, but polynomially solvable for $(k,l)$ -bounded instances for certain $k$ and $l$ . Here, $k$ denotes the maximum number of ambiguous sites in any genotype, and $l$ is the maximum number of genotypes that are ambiguous at the same site. Only the complexity of the $(*,2)$ -bounded problem is still unknown, where $*$ denotes no restriction. It has been proved that $(*,2)$ -bounded instances have compatibility graphs that can be constructed from cliques and circuits by pasting along an edge. In this paper, we give a constructive proof of the fact that $(*,2)$ -bounded instances are polynomially solvable if the compatibility graph is constructed by pasting cliques, trees and circuits along a bounded number of edges. We obtain this proof by solving a slightly generalized problem on circuits, trees and cliques respectively, and arguing that all possible combinations of optimal solutions for these graphs that are pasted along a bounded number of edges can be enumerated efficiently. Keywords: Drug design, hereditary disease, health, haplotyping, graph, polynomial time algorithm, shortest path, matching problem, path partition