Decision Tree Complexity versus Block Sensitivity and Degree

TL;DR

This paper investigates the tightness of cubic upper bounds relating decision tree complexity to block sensitivity and degree, providing improved bounds for specific classes of Boolean functions and exploring their implications in communication complexity.

Contribution

The paper improves cubic upper bounds to quadratic bounds for certain classes of Boolean functions and introduces zebra functions, linking decision tree complexity to communication complexity.

Findings

Quadratic bounds for graph properties and functions with constant alternations.

Introduction of zebra functions with decision tree complexity at most the square of block sensitivity.

Equivalence between bounding decision tree complexity and communication complexity measures.

Abstract

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, both of the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0^n to 1^n has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree. Finally, we show using a lifting theorem of communication complexity by G{\"{o}}{\"{o}}s, Pitassi and Watson that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols.