Computing Graph Properties by Randomized Subcube Partitions

We prove a new lower bound on the randomized decision tree complexity of monotone graph properties. For a monotone graph property $A$ of graphs on $n$ vertices, let $p=p(A)$ denote the threshold probability of $A$, namely the value of $p$ for which a random graph from $G(n,p)$ has property $A$ with probability $1/2$. Then the expected number of queries made by any decision tree for $A$ on such a random graph is at least $\omega(n^2/max{pn, log n})$. Our lower bound holds in the subcube partition model, which generalizes the decision tree model. The proof combines a simple combinatorial lemma on subcube partitions (which may be of independent interest) with simple graph packing arguments. Our approach motivates the study of packing of ``typical’’ graphs, which may yield better lower bounds.