Covering Pairs in Directed Acyclic Graphs
The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this paper, we study the computational complexity of two constrained variants of Minimum Path Cover motivated by the recent introduction of next-generation sequencing technologies in bioinformatics. The first problem (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum cardinality set of paths “covering” all the vertices such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The second problem (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a single path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W-hardness when parametrized by the number of covered pairs. On the positive side, we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree, a natural parameter in the bioinformatics applications of the problem.
Unable to display preview. Download preview PDF.
- 3.Downey, R., Fellows, M.: Parameterized complexity. Springer (1999)Google Scholar
- 5.Eriksson, N., Pachter, L., Mitsuya, Y., Rhee, S., Wang, C., Gharizadeh, B., Ronaghi, M., Shafer, R., Beerenwinkel, N.: Viral population estimation using pyrosequencing. PLoS Computational Biology 4(5), e1000074 (2008)Google Scholar
- 8.Garey, M., Johnson, D.: Computer and intractability: A guide to the theory of NP-completeness. W. H. Freeman (1979)Google Scholar
- 9.Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Univ. Press (2006)Google Scholar
- 11.Trapnell, C., Williams, B., Pertea, G., Mortazavi, A., Kwan, G., van Baren, M., Salzberg, S., Wold, B., Pachter, L.: Transcript assembly and quantification by RNA-Seq reveals unannotated transcripts and isoform switching during cell differentiation. Nature Biotechnology 28(5), 516–520 (2010)CrossRefGoogle Scholar