Lower Bounds for Subgraph Isomorphism

We consider the computational complexity 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. We give an overview of unconditional lower bounds for this problem, focusing on circuits and formulas in the AC 0 and monotone settings. 1 Background and preliminaries The subgraph isomorphism problem is the computational task of determining whether a “host” graph H contains a copy of a “pattern” G as a subgraph. When both G and H are given as input, this is a classic NP -complete problem which generalizes both maximum clique and hamiltonian cycle [21]. We refer to the G-subgraph isomorphism problem in the setting whether the pattern G is fixed and H alone is given as input. This family of problems, which includes the k-clique and k-cycle problems for fixed k, are each solvable in polynomial time O(n|V (G)|) by the obvious exhaustive search.1 Faster algorithms are known in many cases. For example, k-cycle has an O(nω) time algorithm where ω < 2.38 is the exponent of matrix multiplication [2], while k-clique is solvable in O(nωdk/3e) time [27]. Additional upper bounds are based on invariants of G and H (see [25] for a survey), such as an O(nk+1) time algorithm for patterns of tree-width k [28]. The focus of this article is lower bounds: impossibility results which show, unconditionally, that the G-subgraph isomorphism problem cannot be solved with insufficient computational resources, such as time or space. Conditionally, it is known that the Exponential Time Hypothesis implies that k-clique requires time nΩ(k) [9] and that G-subgraph isomorphism Throughout this article, asymptotic notation (O(·), Ω(·), etc.), whenever bounding a function of n, hides constants that may depend on G.