On the Saks-Wigderson Conjecture NingNing Peng Mathematical Institute, Tohoku University

Abstract We investigate the deterministic and the randomized decision tree complexities for Boolean function f , denoted D(f ) and R(f ), respectively. The cost of an algorithm for computing f is measured by the number of input variables read to determine the value of f for the worst case. Thus, D(f ) is the minimum (by the best deterministic algorithm) of the maximal costs (over the probability distributions for inputs) for computing f . So, for a read-once function f , D(f ) = n, where n is the number of input variables. On the other hand, R(f ) is the maximum average cost by the best randomized algorithm for computing f . Clearly, n ≤ R(f ) ≤ D(f ) for every Boolean function f . A major open question is how small R(f ) can be with respect to D(f ). It is well known that R(f ) ≥ D(f )0.5 for every Boolean function f . In 1991, Heiman and Wigderson [1] show that R(f ) ≥ D(f )0.51 . Some Boolean function f is know to have R(f ) = Θ(n0.753 ). The well-know Saks-Wigderson [2] Conjecture is the statement that for any Boolean function f , R(f ) = Ω(n0.753 ) where n is the number of inputs. Liu and Tanaka [3] made some important contributions to this problem. We survey their results and related works.

References [1] Rafi Heiman and Avi Wigderson. Randomized vs. deterministic decision tree complexity for read-once Boolean functions. Comput complexity 1 (1991) 311-329. [2] Michael saks and Avi Wigderson. Probabilistic Boolean decision trees and the complexity of Evaluating Game trees. In Proc. 27th IEEE Symp. On Foundations of Computer Science, (1986) 29-38. [3] ChenGuang Liu and Kazuyuki Tanaka. Eigen-distribution on random assignments for game trees. Information Processing Letters 104 (2007) 73-77.

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May 1, 2012