Hardness Condensation for Decision Tree Measures by Restrictions
Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh
TL;DR
The paper investigates hardness condensation for Boolean functions under restricting variables, focusing on decision-tree measures. It proves lossless condensation is impossible for several measures, including $\mathsf{bs}$, $\mathsf{fbs}$, $\mathsf{C}$, and $\mathsf{D}$, via explicit constructions (notably Modified Rubinstein and a cheat-sheet Tribes variant), achieving $|\rho^{-1}(*)|=O(\mathcal{M}(f))$ but reduced complexity $\mathcal{M}(f|_\rho)=O(\mathcal{M}(f)^{2/3})$ in the negative results. The authors complement these with lossy condensation results, showing that for every $f$ there exists a restriction of size $Θ(\mathcal{M}(f))$ such that many measures satisfy $\mathcal{M}(f|_\rho)=Ω(\mathcal{M}(f)^{1/2})$, and provide weaker exponents for randomized and quantum models. They also present a stronger negative result for deterministic query complexity using cheat-sheet constructions, improving prior bounds. Together, these results map the landscape of hardness condensation by restriction, delineating what can and cannot be preserved under restriction for core complexity measures and highlighting key open questions about tight exponents and optimal bounds.
Abstract
For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $Θ(k)$ variables while keeping the complexity preserved at $Θ(k)$? This question, in the context of query complexity, was recently studied by G{ö}{ö}s, Newman, Riazanov and Sokolov (STOC 2024). They showed, among other results, that query complexity can not be condensed losslessly. They asked if complexity measures like block sensitivity or unambiguous certificate complexity can be condensed losslessly? In this work, we show that decision tree measures like block sensitivity and certificate complexity, cannot be condensed losslessly. That is, there exists a Boolean function $f$ such that any restriction of $f$ to $O(\mathcal{M}(f))$ variables has $\mathcal{M}(\cdot)$-complexity at most $\tilde{O}(\mathcal{M}(f)^{2/3})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$. This also improves upon a result of G{ö}{ö}s, Newman, Riazanov and Sokolov (STOC 2024). We also complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function $f$ there exists a restriction of $f$ to $O(\mathcal{M}(f))$ variables such that its $\mathcal{M}(\cdot)$-complexity is at least $Ω(\mathcal{M}(f)^{1/2})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},λ\}$. We also show a slightly weaker positive result for randomized and quantum query complexity.