On the AC0 Complexity of Subgraph Isomorphism

Let P be a fixed graph (hereafter called a “pattern”), and let SUBGRAPH(P) denote the problem of deciding whether a given graph G contains a subgraph isomorphic to P. We are interested in AC0-complexity of this problem, determined by the smallest possible exponent C(P) for which 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 SUBGRAPHcol(P) in which the target graph G is V(P)colored, and the average-case version SUBGRAP_Have(P) under the distribution G(n, n^-θ(P)), where θ(P) is the threshold exponent of P. Defining C_col(P) and Cave(P) analogously to C(P), our main contributions can be summarized as follows. (1) C_col(P) coincides with the tree-width of the pattern P within a logarithmic factor. This shows that the previously known upper bound by Alon, Yuster, Zwick [3] is almost tight. (2) We give a characterization of Cave(P) in purely combinatorial terms within a multiplicative factor of 2. This shows that the lower bound technique of Rossman [21] is essentially tight, for any pattern P whatsoever. (3) We prove that if Q is a minor of P then SUBGRAPH_col(Q) is reducible to SUBGRAPH_col(P) via a linear-size monotone projection. At the same time, we show that there is no monotone projection whatsoever that reduces SUBGRAPH(M₃) to SUBGRAPH(P₃ + M₂) (P₃ is a path on 3 vertices, Mk 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 (worstcase, uncolored) one.