The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs

AbstractIn this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in “yes”-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W[t] for all t∈N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete

Algorithms For Phylogeny Reconstruction In a New Mathematical Model

The evolutionary history of a set of species is represented by a tree called phylogenetic tree or phylogeny. Its structure depends on precise biological assumptions about the evolution of species. Problems related to phylogeny reconstruction (i.e., finding a tree representation of information regarding a set of items) are widely studied in computer science. Most of these problems have found to be NP-hard. Sometimes they can solved polynomially if appropriate restrictions on the structure of the tree are fixed. This paper summarizes the most recent problems and results in phylogeny reconstruction, and introduces an innovative tree model, called Phylogenetic Parsimonious Tree, which is justified by significant biological hypothesis. Using PPT two problems are studied: the existence and the reconstruction of a tree both when sequences of characters and partial order on interspecies distances are given. We rove complexity results that confirm the hardness of this class of problems