Exploiting bounded signal flow for graph orientation based on cause-effect pairs

Background: We consider the following problem: Given an undirected network and a set of sender–receiver pairs, direct all edges such that the maximum number of “signal flows ” defined by the pairs can be routed respecting edge directions. This problem has applications in understanding protein interaction based cell regulation mechanisms. Since this problem is NP-hard, research so far concentrated on polynomial-time approximation algorithms and tractable special cases. Results: We take the viewpoint of parameterized algorithmics and examine several parameters related to the maximum signal flow over vertices or edges. We provide several fixed-parameter tractability results, and in one case a sharp complexity dichotomy between a linear-time solvable case and a slightly more general NP-hard case. We examine the value of these parameters for several real-world network instances. Conclusions: Several biologically relevant special cases of the NP-hard problem can be solved to optimality. In this way, parameterized analysis yields both deeper insight into the computational complexity and practical solving strategies. Background Current technologies [1] like two-hybrid screening ca

On Making Directed Graphs Transitive

We present the first thorough theoretical analysis of the Transitivity Editing problem on digraphs. Herein, the task is to make a given digraph transitive by a minimum number of arc insertions or deletions. Transitivity Editing has recently been identified as important for the detection of hierarchical structure in molecular characteristics of disease. We present a first proof of NP-hardness, which also extends to the restricted cases where the input digraph is acyclic or has maximum degree three. Moreover, we improve previous fixed-parameter algorithms, now achieving a running time of O(2.57 k + n 3) for an n-vertex digraph if k arc modifications are sufficient to make it transitive. By providing an O(k 2)-vertex problem kernel, we also answer an open question from the literature. In case of digraphs with maximum degree d, an O(k ·d)-vertex problem kernel can be shown. We also demonstrate that if the input digraph does not contain “diamonds”, then there is an optimal solution that performs only arc deletions. Our hardness as well as algorithmic results transfer to Transitivity Deletion, where only arc deletions are allowed