On monotone circuits with local oracles and clique lower bounds

Abstract

We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i=y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $\mu \in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k},V_{n,k}\subseteq \{0,1{\}}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $\mu$. 2. For $5\le k(n)\le n^{1/4}$, arbitrary depth, and $\mu \le 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{\Theta (\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained by Alon and Boppana (1987).

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