Improved Lower Bounds on the Randomized Complexity of Graph Properties
We prove a lower bound of Ω(n 4/3 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.
KeywordsDecision tree complexity monotone graph properties randomized complexity randomized algorithms graph packing probabilistic method
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