LOWER BOUNDS FOR SUBGRAPH ISOMORPHISM

We consider the problem of determining whether an Erdős–Rényi random graph contains a subgraph isomorphic to a fixed pattern, such as a clique or cycle of constant size. The computational complexity of this problem is tied to fundamental open questions including P vs. NP and NC1 vs. L. We give an overview of unconditional average-case lower bounds for this problem (and its colored variant) in a few important restricted classes of Boolean circuits. 1 Background and preliminaries The subgraph isomorphism problem is the computational task of determining whether a “host” graph H contains a subgraph isomorphic to a “pattern” graph G. When both G and H are given as input, this is a classic NP-complete problem which generalizes both the and H problems Karp [1972]. We refer to the Gsubgraph isomorphism problem in the setting where the pattern G is fixed and H alone is given as input. As special cases, this includes the kand kproblems when G is a complete graph or cycle of order k. For patterns G of order k, the G-subgraph isomorphism problem is solvable in time O(n) by the obvious exhaustive search.1 This upper bound can be improved toO(n) using any O(n) time algorithm for fast matrix multiplication Nešetřil and Poljak [1985] (the current record has ̨ < 2:38 Le Gall [2014]). Additional upper bounds are tied to structural parameters of G, such as an O(n) time algorithm for patterns G of treewidth w Plehn and Voigt [1990]. (See Marx and Pilipczuk [2014] for a survey on upper bounds.) The author’s work is supported by NSERC and a Sloan Research Fellowship. MSC2010: primary 68Q17; secondary 05C60. 1Throughout this article, asymptotic notation (O( ), Ω( ), etc.), whenever bounding a function of n, hides constants that may depend on G.