On the Complexity of two Problems on Orientations of Mixed Graphs

Interactions between biomolecules within the cell can be modeled by biological networks, i.e. graphs whose vertices are the biomolecules (proteins, genes, metabolites etc.) and whose edges represent their functional relationships. Depending on their nature, the interactions can be undirected (e.g. protein-protein interactions, PPIs) or directed (e.g. protein-DNA interactions, PDIs). A physical network is a network formed by both PPIs and PDIs, and is thus modeled by a mixed graph. External cellular events are transmitted into the nucleus via cascades of activation/deactivation of proteins, that correspond to paths (called signaling pathways) in the physical network from a source protein (cause) to a target protein (effect). There exists experimental methods to identify the cause-effect pairs, but such methods do not provide the signaling pathways. A key challenge is to infer such pathways based on the cause-effect informations. In terms of graph theory, this problem, called MAXIMUM GRAPH ORIENTATION (MGO), is defined as follows: given a mixed graph G and a set P of source-target pairs, find an orientation of G that replaces each (undirected) edge by a single (directed) arc in such a way that there exists a directed path, from s to t, for a maximum number of pairs (s, t) ∈ P . In this work, we consider a variant of MGO, called S-GO, in which we ask whether all the pairs in P can be connected by a directed path. We also introduce a minimization problem, called MIN-DB-GO, in which all the pairs in P must be connected by a directed path, while we allow some edges of G to be doubly oriented (i.e. replaced by two arcs in opposite directions). We investigate the complexity of S-GO and MIN-DB-GO by considering some restrictions on the input instances (such as the maximum degree of G or the cardinality of P). We provide several polynomial-time algorithms, hardness and inapproximability results that together give an extensive picture of tractable and intractable instances for both problems.