Given an edge-weighted undirected hypergraph [Formula: see text] and an even-sized set of vertices [Formula: see text], a T-cut is a partition of [Formula: see text] into two parts [Formula: see text] and [Formula: see text] such that [Formula: see text] is odd. A T-join in [Formula: see text] for [Formula: see text] is a set of hyperedges [Formula: see text] such that for every T-cut [Formula: see text] there is a hyperedge [Formula: see text] intersecting both [Formula: see text] and [Formula: see text]. A directed hypergraph has for every hyperedge exactly one vertex, called the head, and several vertices, that are tails. A directed hypergraph is strongly connected if there exists at least one directed path between any two vertices of the hypergraph, where a directed path is defined to be a sequence of vertices and hyperedges for which each hyperedge has as one of its tails the vertex preceding it and as its head the vertex following it in the sequence. Orienting an undirected hypergraph means choosing a head for each hyperedge. We prove that every edge-weighted undirected hypergraph that admits a strongly connected orientation has a T-join of total weight at most [Formula: see text] times the total weight of all the edges of the hypergraph, and sketch an improvement to [Formula: see text]. We also exhibit a series of example showing that one cannot improve the constant above to [Formula: see text].
Central hubs prediction for Bio Networks by Directed Hypergraph - GA with validation to COVID-19 PPI
