Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

Gurumuhkani et al. (CCC'24) introduced the local enumeration problem $Enum(k, t)$ as follows: for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment with Hamming weight less than $t(n)$, enumerate all satisfying assignments of Hamming weight exactly $t(n)$. They showed that efficient algorithms for local enumeration yield new $k$-SAT algorithms and depth-3 lower bounds for Majority function. As the first non-trivial case, they gave an algorithm for $k = 3$ which in particular gave a new lower bound on the size of depth-3 circuits with bottom fan-in at most 3 computing Majority. In this paper, we give an optimal algorithm that solves local enumeration on monotone formulas for $k = 3$ and all $t \le n/2$. In particular, we obtain an optimal lower bound on the size of monotone depth-3 circuits with bottom fan-in at most 3 computing Majority.

💡 Research Summary

The paper studies the complexity of monotone depth‑3 circuits with bottom fan‑in three (Σ₃³ circuits) for computing the Majority function. Building on the “local enumeration” framework introduced by Gurumuhkani et al. (CCC’24), the authors focus on the problem Enum⁺(k, t): given a monotone k‑CNF formula that has no satisfying assignment of Hamming weight smaller than t, enumerate all satisfying assignments of exact weight t. They show that an efficient algorithm for Enum⁺(k, t) yields (i) a non‑trivial SAT algorithm with savings, and (ii) a quantitative lower bound on the size of Σₖ³ circuits computing Majority.

The main technical contribution is an optimal deterministic algorithm for Enum⁺(3, t) for every t ≤ n/2, running in time poly(n)·6^{n/4}. To achieve this, the authors first define Θ(n, t, k) as the maximum number of t‑transversals (i.e., weight‑t satisfying assignments) that any monotone k‑CNF on n variables can have while being consistent with a given Boolean function. A t‑transversal is a set of variables intersecting every clause; for monotone formulas this coincides with a satisfying assignment of weight t. The quantity Θ(n, t, 3) therefore captures the combinatorial bottleneck for Enum⁺(3, t).

The paper constructs extremal monotone 3‑CNFs that achieve Θ(n, t, 3). Two basic building blocks are introduced: (1) cliques K_{3,ℓ}, the set of all 3‑clauses on ℓ variables, which are extremal for thresholds ℓ‑2; (2) Turán‑style constructions T_{3,n}, obtained by partitioning the variables into three almost‑equal parts and adding all intra‑part and adjacent‑part 3‑clauses. The former yields Θ(ℓ, ℓ‑2, 3)=C(ℓ, ℓ‑2), while the latter captures the case t = n‑3 and connects the problem to the classical hypergraph Turán problem.

For the range 0 ≤ t ≤ ⌊n/3⌋ the extremal construction is simply S_{3,n,t}=t·K_{3,3}, i.e., t disjoint copies of the 3‑variable clique. For t close to n (specifically n‑k+i with 1 ≤ i ≤ k) the extremal family consists of all (k‑i+1)‑clauses, a straightforward generalisation. The challenging regime is n/3 < t ≤ n/2. Here the authors combine cliques and Turán blocks, carefully arranging them into blocks of four variables. Within each block there are exactly six minimal transversals, and the blocks are independent. By an inductive argument that respects the “critical clause” property (every variable in a minimal transversal belongs to a unique clause intersecting the transversal only at that variable), they prove that no monotone 3‑CNF can have more than 6·⌊n/4⌋ transversals of weight ⌊n/2⌋. This matches the construction, establishing Θ(n, ⌊n/2⌋, 3)=6·⌊n/4⌋.

The enumeration algorithm exploits this block structure: it enumerates all 6^{n/4} possible choices of a transversal within each block, concatenates them, and verifies satisfaction (which is guaranteed by construction). The runtime is therefore poly(n)·6^{n/4}, which is optimal because any faster algorithm would contradict the proven extremal bound.

Finally, the authors translate the combinatorial result into circuit lower bounds via a simple counting argument (Fact 2.11). Any monotone Σ₃³ circuit computing Majority must have top‑OR fan‑in at least |Maj^{-1}(1)| / Θ(n, n/2, 3). Since |Maj^{-1}(1)| = Θ(2^{n} / √n) and Θ(n, n/2, 3)=6·⌊n/4⌋, the lower bound simplifies to (2 / 6^{1/4})·n − o(n) ≈ 1.277 n. This matches the best known upper bound (also (2 / 6^{1/4})·n up to lower‑order terms) and thus settles the exact asymptotic size of monotone depth‑3 circuits with bottom fan‑in three for Majority.

The paper concludes by discussing the broader implications: the method of constructing extremal CNFs guided by upper‑bound intuition may apply to other threshold functions, and the tight connection between local enumeration and circuit lower bounds suggests a pathway to attack the long‑standing depth‑3 barrier for non‑monotone circuits. The authors also note that extending the approach to larger bottom fan‑in k or to non‑monotone settings remains an open and promising direction.