Exhaustive Search Is Hard to Beat in Constant Depth for Hyperclique and Orthogonal Vectors

Abstract

A hyperclique in an $r$-uniform hypergraph is a set of vertices in which every $r$-subset is an edge. A simple exhaustive search algorithm detects a hyperclique on $\ell$ vertices in an $r$-uniform hypergraph on $m$ vertices by checking all $\binom{m}{\ell}$ sets of $\ell$ vertices. This algorithm can be expressed as a depth-two circuit with at most $m^\ell$ gates and wires. For hypercliques of *constant* size, Amano (2010) showed that this circuit is almost optimal: for large $r$, constant-depth circuits need $m^{\ell(1-O(\log r/r))}$ wires. We prove that this circuit remains hard to beat even for hypercliques of *polynomial* size. More precisely, circuits of depth $h$ need $m^{Ω(\ell)}$ gates for $\ell\leq m^{1/h-\varepsilon}$, and $m^{\ell(1-O(\log r/r))}$ gates for $\ell=m^{o(1)}$ that tends to infinity. Our hyperclique bound and the method behind it also lead to lower bounds for two central problems of fine-grained complexity. First, we extend the bound $n^{k/4}$ of Rossman (2008) for $k$-Clique from cliques of *constant* size to cliques of *polynomial* size: circuits of depth $h$ detecting a clique on $k$ vertices in an $n$-vertex graph need $n^{Ω(k)}$ gates for $k\leq n^{1/(h-1)-\varepsilon}$. Second, a reduction of Choudhury, Limaye, Sreenivasaiah, and Srinivasan (2026) turns our hyperclique bound into a lower bound of $n^{k-\varepsilon}$ gates for $k$-Orthogonal Vectors ($k$-OV) at every fixed depth for vectors of dimension $\log^{O(1)} n$; this proves the $k$-OV Conjecture for constant-depth circuits in polylogarithmic dimension. Finally, for $2$-OV in logarithmic dimension, we give a simple proof that depth-three circuits need $Ω(n^2)$ wires, which answers a question of Kane and Williams (2019).