Improved lower bounds on the randomized complexity of graph properties

Abstract

We prove a lower bound of ω (n4/3 log1/3 n) on the randomized decision tree complexity of any nontrivial monotone n-vertex bipartite graph property, thereby improving the previous bound of ω(n 4/3) due to Hajnal [H91]. Our proof works by improving a probabilistic argument in that paper, which also improves a graph packing lemma proved there. By a result of Gröger [G92] our complexity lower bound carries over from bipartite to general monotone n-vertex graph properties. Graph packing being a well-studied subject in its own right, our improved packing lemma and the probabilistic technique used to prove it, may be of independent interest. © 2011 Springer-Verlag Berlin Heidelberg.