Hardness of clique approximation for monotone circuits

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 Erd\H{o}s-Renyi graph $\mathcal{G}_{n,p}$, with $p=n^{-\frac{2}{\alpha-1}}$ chosen st. with high probability it does not even have an $\alpha$-clique, from a random clique on $\beta$ vertices (where $\alpha \leq \beta$). Using the approximation method of Razborov, Alon and Boppana showed in 1987 that as long as $\sqrt{\alpha} \beta < n^{1-\delta}/\log n$, this problem requires a monotone circuit of size $n^{\Omega(\delta\sqrt{\alpha})}$, implying a lower bound of $2^{\tilde\Omega(n^{1/3})}$ 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 $n^{\Omega(k)}$, in a limited range $k \leq n^{1/3}$, implying a comparable $2^{\tilde{\Omega}(n^{1/3})}$ 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 $\alpha \beta < n^{1-\delta}/\log n$, any monotone circuit rejecting $\mathcal{G}_{n,p}$ while accepting a $\beta$-clique needs to have size at least $n^{\Omega(\delta^2 \alpha)}$; this implies a stronger $2^{\tilde{\Omega}(\sqrt{n})}$ lower bound for the unrestricted version of the problem. We complement this result with a construction of an explicit monotone circuit of size $O(n^{\delta^2 \alpha/2})$ which rejects $\mathcal{G}_{n,p}$, and accepts any graph containing $\beta$-clique whenever $\beta > n^{1-\delta}$. Those two theorems explain the largest $\beta$-clique that can be distinguished from $\mathcal{G}_{n, 1/2}$: when $\beta > n / 2^{C \sqrt{\log n}}$, polynomial size circuit co do it, while for $\beta < n / 2^{\omega(\sqrt{\log n})}$ every circuit needs size $n^{\omega(1)}$.