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

TL;DR

This paper explores the relationship between biclique coverings and rectifier networks in bipartite graphs, demonstrating significant complexity gaps and implications for automata with epsilon transitions, with potential impacts on set cover problems.

Contribution

It establishes a superlinear separation between biclique covering number and rectifier network size, and applies this to automata transformation complexity.

Findings

Existence of graphs with ov(G) rac{3}{2}-\u03b5} e2a4a4 rect(G)

Automata with psilon-transitions requiring ig-O(n^{3/2-psilon}) states when epsilon-free versions are considered

Connections to weighted set cover problem bounds

Abstract

We relate two complexity notions of bipartite graphs: the minimal weight biclique covering number $\mathrm{Cov}(G)$ and the minimal rectifier network size $\mathrm{Rect}(G)$ of a bipartite graph $G$. We show that there exist graphs with $\mathrm{Cov}(G)\geq \mathrm{Rect}(G)^{3/2-\epsilon}$. As a corollary, we establish that there exist nondeterministic finite automata (NFAs) with $\varepsilon$-transitions, having $n$ transitions total such that the smallest equivalent $\varepsilon$-free NFA has $\Omega(n^{3/2-\epsilon})$ 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.