Biclique coverings, rectifier networks and the cost of $\varepsilon$-removal

We relate two complexity notions of bipartite graphs: the minimal weight biclique covering number Cov(G) and the minimal rectifier network size Rect(G) of a bipartite graph G. We show that there exist graphs with Cov(G) ≥ Rect(G). As a corollary, we establish that there exist nondeterministic finite automata (NFAs) with εtransitions, having n transitions total such that the smallest equivalent ε-free NFA has Ω(n) transitions. We also formulate a version of previous bounds for the weighted set cover problem and discuss its connections to giving upper bounds for the possible blow-up.