Hardness of clique approximation for monotone circuits
Jarosław Błasiok, Linus Meierhöfer
TL;DR
This work studies the hardness of approximating clique size within monotone circuits by distinguishing a random graph G_{n,p} from a β-clique, leveraging the Razborov–Alon–Boppana approximation framework together with the latest sunflower bounds. The authors introduce a strengthened distributional problem D(𝒯^-_α, 𝒯^+_β) and prove lower bounds of size n^{Ω(δ^2 α)} when αβ < n^{1-δ}/log n, while also constructing explicit upper bounds of size O(n^{δ^2 α/2}) for β ≥ n^{1-δ}; these results translate into near-tight bounds for the standard Clique problem, giving 2^{˜Ω(√n)} lower bounds and illuminating a polynomial vs. superpolynomial transition around β ≈ n/2^{√{log n}}. Central to the approach is the robust sunflower framework: robust clique sunflowers and the associated RB and RCB numbers, with recent sunflower breakthroughs (Alweiss–Lovett–Wu–Zhang and successors) enabling stronger bounds that surpass prior Alon–Boppana performance. The paper also provides an explicit monotone circuit achieving the upper bound in the favorable regime, demonstrating a clear dichotomy between efficient and intractable regimes and advancing the understanding of monotone circuit complexity for clique-like problems.
Abstract
We consider a problem of approximating the size of the largest clique in a graph, with a monotone circuit. Concretely, we focus on distinguishing a random Erdős-Renyi graph $\mathcal{G}_{n,p}$, with $p=n^{-\frac{2}{α-1}}$ chosen st. with high probability it does not even have an $α$-clique, from a random clique on $β$ vertices (where $α\leq β$). Using the approximation method of Razborov, Alon and Boppana showed in 1987 that as long as $\sqrtα β< n^{1-δ}/\log n$, this problem requires a monotone circuit of size $n^{Ω(δ\sqrtα)}$, implying a lower bound of $2^{\tildeΩ(n^{1/3})}$ for the exact version of the problem when $k\approx n^{2/3}$. Recently Cavalar, Kumar, and Rossman improved their result by showing the tight lower bound $n^{Ω(k)}$, in a limited range $k \leq n^{1/3}$, implying a comparable $2^{\tildeΩ(n^{1/3})}$ lower bound. We combine the ideas of Cavalar, Kumar and Rossman with the recent breakthrough results on the sunflower conjecture by Alweiss, Lovett, Wu and Zhang to show that as long as $αβ< n^{1-δ}/\log n$, any monotone circuit rejecting $\mathcal{G}_{n,p}$ while accepting a $β$-clique needs to have size at least $n^{Ω(δ^2 α)}$; this implies a stronger $2^{\tildeΩ(\sqrt{n})}$ lower bound for the unrestricted version of the problem. We complement this result with a construction of an explicit monotone circuit of size $O(n^{δ^2 α/2})$ which rejects $\mathcal{G}_{n,p}$, and accepts any graph containing $β$-clique whenever $β> n^{1-δ}$. Those two theorems explain the largest $β$-clique that can be distinguished from $\mathcal{G}_{n, 1/2}$: when $β> n / 2^{C \sqrt{\log n}}$, polynomial size circuit co do it, while for $β< n / 2^{ω(\sqrt{\log n})}$ every circuit needs size $n^{ω(1)}$.