DNF Formulas Are Efficiently Testable with Relative Error

We give a poly(s,1/ε)-query algorithm for testing whether an unknown and arbitrary function f: {0,1}^n →{0,1} is an s-term DNF, in the challenging relative-error framework for Boolean function property testing that was recently introduced and studied in a number of works [CDH+25b, CPPS25a, CPPS25b, CDH+25a]. This gives the first example of a rich and natural class of functions which may depend on a super-constant number of variables and yet is efficiently testable in the relative-error model with constant query complexity. A crucial new ingredient enabling our approach is a novel decomposition of any s-term DNF formula into “local clusters” of terms. Our results demonstrate that this new decomposition can be usefully exploited for algorithms even when the s-term DNF is not explicitly given; we believe that this decomposition may have applications in other contexts.