On Condensation of Block Sensitivity, Certificate Complexity and the $\mathsf{AND}$ (and $\mathsf{OR}$) Decision Tree Complexity
This addresses a theoretical problem in computational complexity for researchers, providing counterexamples that clarify the condensability of key complexity measures, though it is incremental in nature.
The paper tackled the hardness condensation question for Boolean functions, showing that block sensitivity, certificate complexity, and AND/OR decision tree complexity do not condense under restrictions, with a specific function achieving only O(k^{2/3}) block sensitivity when restricted to O(k) variables, answering an open question negatively.
Given an $n$-bit Boolean function with a complexity measure (such as block sensitivity, query complexity, etc.) $M(f) = k$, the hardness condensation question asks whether $f$ can be restricted to $O(k)$ variables such that the complexity measure is $Ω(k)$? In this work, we study the condensability of block sensitivity, certificate complexity, AND (and OR) query complexity and Fourier sparsity. We show that block sensitivity does not condense under restrictions, unlike sensitivity: there exists a Boolean function $f$ with query complexity $k$ such that any restriction of $f$ to $O(k)$ variables has block sensitivity $O(k^{\frac{2}{3}})$. This answers an open question in Göös, Newman, Riazanov, and Sokolov (2024) in the negative. The same function yields an analogous incondensable result for certificate complexity. We further show that $\mathsf{AND}$(and $\mathsf{OR}$) decision trees are also incondensable.