文章基本信息

标题：On monotone circuits with local oracles and clique lower bounds
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作者：Jan Krajíček ; Igor C. Oliveira
期刊名称：Chicago Journal of Theoretical Computer Science
印刷版ISSN：1073-0486
出版年度：2018
卷号：2018
出版社：MIT Press ; University of Chicago, Department of Computer Science
摘要：
We investigate monotone circuits with local oracles [Krajíček 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.
- (i) 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 $\mu$.
- (ii) For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $\mu \leq 1/50$, 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 in [Alon and Boppana, 1987].