On the $AC^0$ Complexity of Subgraph Isomorphism

Abstract

Let $P$ be a fixed graph (hereafter called a “pattern''), and let ${\sc Subgraph}(P)$ denote the problem of deciding whether a given graph $G$ contains a subgraph isomorphic to $P$. We are interested in $AC^0$-complexity of this problem, determined by the smallest possible exponent $C(P)$ for which ${\sc Subgraph}(P)$ possesses bounded-depth circuits of size $n^{C(P)+o(1)}$. Motivated by the previous research in the area, we also consider its “colorful” version ${\sc Subgraph}_\mathsf{col}(P)$ in which the target graph $G$ is $V(P)$-colored, and the average-case version ${\sc Subgraph}_\mathsf{ave}(P)$ under the distribution $G(n,n^{-\theta(P)})$, where $\theta(P)$ is the threshold exponent of $P$. Defining $C_\mathsf{col}(P)$ and $C_\mathsf{ave}(P)$ analogously to $C(P)$, our main contributions can be summarized as follows: (1) $C_\mathsf{col}(P)$ coincides with the treewidth of the pattern $P$ up to a logarithmic factor. This shows that the previously known upper bound by Alon, Yuster, and Zwick [J. ACM, 42 (1995), pp. 844--856] is almost tight. (2) We give a characterization of $C_\mathsf{ave}(P)$ in purely combinatorial terms up to a multiplicative factor of 2. This shows that the lower bound technique of Rossman [Proceedings of the 40th ACM Symposium on Theory of Computing, 2008, pp. 721--730] is essentially tight for any pattern $P$ whatsoever. (3) We prove that if $Q$ is a minor of $P$, then ${\sc Subgraph}_\mathsf{col}(Q)$ is reducible to ${\sc Subgraph}_\mathsf{col}(P)$ via a linear-size monotone projection. At the same time, we show that there is no monotone projection whatsoever that reduces ${\sc Subgraph}(M_3)$ to ${\sc Subgraph}(P_3 + M_2)$ ($P_3$ is a path on three vertices, $M_k$ is a matching with $k$ edges, and “+” stands for the disjoint union). This result strongly suggests that the colorful version of the subgraph isomorphism problem is much better structured and well-behaved than the standard (worst-case, uncolored) one.