DNF formulas are efficiently testable with relative error

Xi Chen; William Pires; Toniann Pitassi; Rocco A. Servedio

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arXiv · 2601.16076

DNF formulas are efficiently testable with relative error

Xi Chen·William Pires·Toniann Pitassi·Rocco A. Servedio

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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.

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Xi Chen, William Pires, Toniann Pitassi, Rocco A. Servedio. 2026-01-22. DNF formulas are efficiently testable with relative error. https://arxiv.org/abs/2601.16076

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