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arXiv · 1406.0017

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

Szabolcs Iván·Ádám Dániel Lelkes·Judit Nagy-György·Balázs Szörényi·György Turán
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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 $\mathrm{Cov}(G)\geq \mathrm{Rect}(G)^{3/2-ε}$. 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 $Ω(n^{3/2-ε})$ 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.

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Szabolcs Iván, Ádám Dániel Lelkes, Judit Nagy-György, Balázs Szörényi, György Turán. 2014-05-30. Biclique coverings, rectifier networks and the cost of $\varepsilon$-removal. https://arxiv.org/abs/1406.0017

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