Testing Juntas and Junta Subclasses with Relative Error

Abstract: This papers considers the junta testing problem in a recently introduced ``relative error'' variant of the standard Boolean function property testing model. In relative-error testing we measure the distance from $f$ to $g$, where $f,g: {0,1}ⁿ \to {0,1}$, by the ratio of $|f^{-1}(1) \triangle g<sup>{-1}(1)|$ (the number of inputs on which $f$ and $g$ disagree) to $|f<sup>{-1}(1)|$ (the number of satisfying assignments of $f$), and we give the testing algorithm both black-box access to $f$ and also access to independent uniform samples from $f<sup>{-1}(1)$. Chen et al. (SODA 2025) observed that the class of $k$-juntas is $\text{poly}(2<sup>k,1/\epsilon)$-query testable in the relative-error model, and asked whether $\text{poly}(k,1/\epsilon)$ queries is achievable. We answer this question affirmatively by giving a $\tilde{O}(k/\epsilon)$-query algorithm, matching the optimal complexity achieved in the less challenging standard model. Moreover, as our main result, we show that any subclass of $k$-juntas that is closed under permuting variables is relative-error testable with a similar complexity. This gives highly efficient relative-error testing algorithms for a number of well-studied function classes, including size-$k$ decision trees, size-$k$ branching programs, and size-$k$ Boolean formulas.