The relative-error property testing model was introduced in [Chen et al., 2024] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the function. In this framework, the distance from the unknown target function f that is being tested to a function g is defined as Vol(f△g)/Vol(f), where the numerator is the fraction of inputs on which f and g disagree and the denominator is the fraction of inputs that satisfy f. Recent work [Chen et al., 2026] has shown that over the Boolean domain {0,1}ⁿ, any relative-error testing algorithm for the fundamental class of {halfspaces} (i.e. linear threshold functions) must make Ω(log n) oracle calls. In this paper we complement the [Chen et al., 2026] lower bound by showing that halfspaces can be relative-error tested over ℝⁿ under the standard N(0,I_n) Gaussian distribution using a sublinear number of oracle calls - in particular, substantially fewer than would be required for learning. Our results use a wide range of tools including Hermite analysis, Gaussian isoperimetric inequalities, and geometric results on noise sensitivity and surface area.

Several recent works (Chen et al., SODA 2025; Chen et al., ICALP 2025; Chen et al., COLT 2025; Chen et al., manuscript) have studied a model of property testing of Boolean functions under a relative-error criterion. In this model, the distance from a target function \(f : \{0, 1\}^n \rightarrow \{0, 1\}\) that is being tested to a function \(g\) is defined relative to the number of inputs \(x\) for which \(f(x) = 1\); moreover, testing algorithms in this model have access both to a black-box oracle for \(f\) and to independent uniform satisfying assignments of \(f\). The motivation for this model is that it provides a natural framework for testing sparse Boolean functions that have few satisfying assignments, analogous to well-studied models for property testing of sparse graphs.

We consider the relative abilities and limitations of computationally efficient algorithms for learning in the presence of noise, under two well-studied and challenging adversarial noise models for learning Boolean functions: malicious noise, in which an adversary can arbitrarily corrupt a random subset of examples given to the learner; and nasty noise, in which an adversary can arbitrarily corrupt an adversarially chosen subset of examples given to the learner.

The standard model of Boolean function property testing is not well suited for testing sparse functions which have few satisfying assignments, since every such function is close (in the usual Hamming distance metric) to the constant-0 function. In this work we propose and investigate a new model for property testing of Boolean functions, called relative-error testing, which provides a natural framework for testing sparse functions.

Abstract. It is a longstanding open problem whether the Minimum Circuit Size Problem ([Formula: see text]) and related meta-complexity problems are [Formula: see text]-complete and hard to approximate. In this work, we prove NP-hardness of approximating meta-complexity with nearly optimal approximation gaps. Our key idea is to use cryptographic constructions in our reductions, where the security of the cryptographic construction implies the correctness of the reduction. We present three results that give both conditional and unconditional hardness of approximation. First, assuming subexponentially-secure witness encryption exists, we prove essentially optimal NP-hardness of approximating conditional time-bounded Kolmogorov complexity ([Formula: see text]) in the regime where [Formula: see text]. Second, we unconditionally show near-optimal NP-hardness of approximation for the minimum oracle circuit size problem where Yes instances have circuit complexity at most [Formula: see text], and No instances are essentially as hard as random truth tables. Finally, we define a “multivalued” version of [Formula: see text], called [Formula: see text], and show that with probability 1 over a random oracle [Formula: see text], [Formula: see text] is NP-hard to approximate under quasi-polynomial-time reductions with [Formula: see text] oracle access.

The range avoidance problem, denoted as C-Avoid, asks to find a non-output of a given C-circuit C:0,1^n -> 0,1^l with stretch l>n. This problem has recently received much attention in complexity theory for its connections with circuit lower bounds and other explicit construction problems. Inspired by the Algorithmic Method for circuit lower bounds, Ren, Santhanam, and Wang (FOCS’22) established a framework to design FP^NP algorithms for C-Avoid via slightly non-trivial data structures related to C. However, a major drawback of their approach is the lack of unconditional results even for C=AC^0.

It is a long-standing open problem whether the Minimum Circuit Size Problem (MCSP) and related meta-complexity problems are NP-complete. Even for the rare cases where the NP-hardness of meta-complexity problems are known, we only know very weak hardness of approximation.