On the power of randomness in the decision tree model

Results suggest that there are relations between the decision tree complexity of a Boolean function and its symmetry. A central conjecture is that for any monotone graph property the randomized decision tree complexity does not differ from the deterministic one with more than a constant factor. The authors improve on V. King´s Ω(n^5/4) lower bound on the randomized decision tree complexity of monotone graph properties to Ω(n^4/3). The proof follows A. Yao´s (1977) approach and improves it in a different direction from King´s. At the heart of the proof is a duality argument combined with a new packing lemma for bipartite graphs. Consideration is also given to the question of what distinguishes graph properties from other highly symmetric Boolean functions, where randomization can help significantly. Open questions concerning this problem are discussed