DNF formulas are efficiently testable with relative error

TL;DR

This paper presents an efficient property testing algorithm for s-term DNF formulas under the relative-error framework, introducing a novel decomposition technique that enables testing with constant query complexity.

Contribution

It introduces the first efficient testing algorithm for s-term DNF formulas in the relative-error model, utilizing a new decomposition into local clusters of terms.

Findings

Poly$(s,1/psilon)$ query complexity for testing DNF formulas.

First example of a natural class of functions with super-constant variables that is efficiently testable in this model.

Decomposition into local clusters can be exploited for algorithms even without explicit formula access.

Abstract

We give a poly$(s,1/\epsilon)$-query algorithm for testing whether an unknown and arbitrary function $f: \{0,1\}^n \to \{0,1\}$ 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.