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Decision Tree Complexity Versus Block Sensitivity and Degree

Authors Rahul Chugh, Supartha Podder, Swagato Sanyal



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Author Details

Rahul Chugh
  • Citadel Securities, London, UK
Supartha Podder
  • Stony Brook University, NY, USA
Swagato Sanyal
  • IIT Kharagpur, India

Acknowledgements

We thank anonymous referees for many helpful feedback.

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Rahul Chugh, Supartha Podder, and Swagato Sanyal. Decision Tree Complexity Versus Block Sensitivity and Degree. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 27:1-27:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.27

Abstract

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. While decision tree complexity is long known to be polynomially related with many other measures, the optimal exponents of many of these relations are not known. 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. Proving quadratic relations between these measures would resolve several open questions in decision tree complexity. For example, it will imply a tight relation between decision tree complexity and square of randomized decision tree complexity and a tight relation between zero-error randomized decision tree complexity and square of fractional block sensitivity, resolving an open question raised by Aaronson [Aaronson, 2008]. 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, 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 0ⁿ to 1ⁿ 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 Göös, Pitassi and Watson [Göös et al., 2017] 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
Keywords
  • Query complexity
  • Graph Property
  • Boolean functions

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