Bounding the randomized decision tree complexity of read-once Boolean functions

Abstract

We investigate the deterministic and the randomized decision tree complexities of Boolean functions, denoted by D(f) and R(f), respectively. A long standing conjecture is that, for every Boolean function f, R(f) = Ω (D (f)α) where α = log2 (1+√33/4) = 0.753 ... [Saks-Wigderson, FOCS'86]. In this paper, we concentrate on the class of read-once Boolean functions and propose a promising approach to attack the conjecture for this class. Precisely, we give a statement about a property of a real-valued function whose correctness implies the conjecture for all read-once Boolean functions. So far we have not succeeded to prove this statement; however, we verified by computer calculation that the statement is "at least approximately true" that implies a lower bound of R(f) = Ω(D(f)0.99α) = Ω(D(f)0.746). This improves the best known lower bound of Ω(D(f )0.51) by Heiman and Wigderson [Comput. Complexity, 1991].