Decision-tree complexity vs. approximate nondeterministic degree

Prove that for every Boolean function f: {0,1}^n → {0,1} and constant ε ∈ [0,1), the decision-tree complexity satisfies D(f) ≤ O(ndeg_ε(f)^2 · ndeg_ε(¬f)^2), where ndeg_ε(·) denotes ε-approximate nondeterministic degree and ¬f is the Boolean negation of f.

Background

The authors define ε-approximate nondeterministic degree and show partial results: for constant ε they prove D(f) ≤ O(ndeg_ε(f)2 * ndeg(¬f)2) and D(f) ≤ O(ndeg(f)2 * ndeg_ε(¬f)2).

They conjecture that adapting the combinatorial part of their proof would yield a bound involving approximate nondeterministic degree on both sides.

References

If we could similarly adapt the combinatorial side of our proof, namely \cref{lem:hitset_upper}, then we would prove the following conjecture. \begin{conjecture} For every Boolean function $f$, and constant $\epsilon\in [0,1)$, \begin{equation} D(f)\leq O(\ndeg_\epsilon(f)2\ndeg_\epsilon(\neg f)2). \end{equation} \end{conjecture}

— Rational degree is polynomially related to degree  (2601.08727 - Kovacs-Deak et al., 13 Jan 2026) in Section 4.3