Relations between monotone complexity measures based on decision tree complexity
Farzan Byramji, Vatsal Jha, Chandrima Kayal, Rajat Mittal
TL;DR
This work investigates monotone analogues of key Boolean complexity measures—$\mathsf{mbs}(f)$, $\mathsf{fmbs}(f)$, and $\mathsf{MCC}(f)$—and their connections to AND-decision trees and sparsity. It proves that for any $f$ the ratio $\frac{\mathsf{fmbs}(f^{\ell})}{\mathsf{mbs}(f^{\ell})}$ is bounded by a function of the input size $n$ independent of the composition depth $\ell$, enabling a lifting of upper bounds from monotone block sensitivity to its fractional analogue. The paper also shows that tightening the bound of $\mathsf{mbs}(f)$ in terms of $\log(\spar(f))$ is equivalent to tightening the bound of $\mathsf{s}(f)$ in terms of $\deg(f)$, linking two central questions in the field; for monotone and symmetric functions, it establishes that $\mathsf{MCC}(f)=\mathsf{ms}(f)=\mathsf{mbs}(f)=\mathsf{fmbs}(f)$, yielding improved general bounds $\mathsf{MCC}(f)=O(\log^{2}(\spar(f)))$ and, for symmetric functions, $\mathsf{MCC}(f)=(1+o(1))\log(\spar(f))$. The work also notes limitations of certain lifting approaches for $\log(\spar(f))$ and poses open questions about the tightness of these relationships and separations between monotone measures.
Abstract
In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to log n factor, for any Boolean function composed with AND function as the inner gadget. One of the main tools in this result was the relationship between monotone analogues of well-studied Boolean complexity measures like block sensitivity and certificate complexity. The relationship between the standard measures has been a long line of research, with a landmark result by Huang (Annals of Mathematics 2019), finally showing that sensitivity is polynomially related to all other standard measures. In this article, we study the monotone analogues of standard measures like block sensitivity (mbs(f)), certificate complexity (MCC(f)) and fractional block sensitivity (fmbs(f)); and study the relationship between these measures given their connection with AND-decision tree and sparsity of a Boolean function. We show the following results: 1) Given a Boolean function $f : \{0, 1\}^{n} \rightarrow \{0, 1\}$, the ratio $fmbs(f^l )/mbs(f^l )$ is bounded by a function of n (and not l). A similar result was known for the corresponding standard measures (Tal, ITCS 2013). This result allows us to extend any upper bound by a well behaved measure on monotone block sensitivity to monotone fractional block sensitivity. 2) The question of the best possible upper bound on monotone block sensitivity by the logarithm of sparsity is equivalent to the natural question of best upper bound by degree on sensitivity. One side of this relationship was used in the proof by Knop, Lovett, McGuire and Yuan (STOC 2021). 3) For two natural classes of functions, symmetric and monotone, hitting set complexity (MCC) is equal to monotone sensitivity.