Hardness of Clique Approximation for Monotone Circuits
Błasiok, Jarosław, Meierhöfer, Linus · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2025
We consider a problem of approximating the size of the largest clique in a graph, using a monotone circuit. Concretely, we focus on distinguishing a random Erdős–Rényi graph 𝒢_{n,p}, with p = n^{-2/(α-1)} chosen st. with high probability it does not even contain an α-clique, from a random clique on β vertices (where α ≤ β). Using the approximation method of Razborov, Alon and Boppana showed in their influential work in 1987 that as long as √{α} β n^{1-δ}. In particular, those two theorems give a precise characterization of the smallest β-clique that can be distinguished from 𝒢_{n, 1/2}: when β > n / 2^{C √{log n}}, there is a polynomial-size circuit that solves it, while for β < n / 2^ω(√{log n}) every circuit needs size n^ω(1).