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 path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W[1]-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

Vertex Disjoint Path in Upward Planar Graphs

The $k$-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed $k$ when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter $k$) on this restricted class. Only recently, \cite{CMPP}.\ achieved a major breakthrough and answered the question positively. Despite the importance of this result (and the brilliance of their proof), it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has at least double exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller class of planar digraphs, the class of upward planar digraphs, a well studied class of planar graphs which can be drawn in a plane such that all edges are drawn upwards. We show that on the class of upward planar digraphs the problem (i) remains NP-complete and (ii) the problem is fixed-parameter tractable. While membership in FPT follows immediately from \cite{CMPP}'s general result, our algorithm has only single exponential parameter dependency compared to the double exponential parameter dependence for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in \cite{CMPP}.Comment: 14 page

On the fixed-parameter tractability of the maximum connectivity improvement problem

In the Maximum Connectivity Improvement (MCI) problem, we are given a directed graph $G=(V,E)$ and an integer $B$ and we are asked to find $B$ new edges to be added to $G$ in order to maximize the number of connected pairs of vertices in the resulting graph. The MCI problem has been studied from the approximation point of view. In this paper, we approach it from the parameterized complexity perspective in the case of directed acyclic graphs. We show several hardness and algorithmic results with respect to different natural parameters. Our main result is that the problem is $W[2]$-hard for parameter $B$ and it is FPT for parameters $|V| - B$ and $\nu$, the matching number of $G$. We further characterize the MCI problem with respect to other complementary parameters.Comment: 15 pages, 1 figur