Current Algorithms for Detecting Subgraphs of Bounded Treewidth Are Probably Optimal

The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time O(n^{tw(H)+1}) [Alon, Yuster, Zwick\u2795], where n is the number of vertices of the host graph G. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of O(n^{tw(H)+1-?}) or even faster (e.g. for k-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time n^{o(tw(H) / log(tw(H)))} for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx\u2707]. In this paper, we demonstrate the existence of maximally hard pattern graphs H that require time n^{tw(H)+1-o(1)}. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth t: For any ? > 0 there exists t ? 3 and a pattern graph H of treewidth t such that Subgraph Isomorphism on pattern H has no algorithm running in time O(n^{t+1-?}). Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth t ? 3: For any t ? 3 there exists a pattern graph H of treewidth t such that for any ? > 0 Subgraph Isomorphism on pattern H has no algorithm running in time O(n^{t+1-?}). In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for tw < 3, (3) Subgraph Isomorphism parameterized by pathwidth instead of treewidth, and (4) a weighted variant that we call Exact Weight Subgraph Isomorphism, for which we examine pseudo-polynomial time algorithms. For many of these settings we obtain similarly tight upper and lower bounds

On the First-Order Complexity of Induced Subgraph Isomorphism

Given a graph F, let I(F) be the class of graphs containing F as an induced subgraph. Let W[F] denote the minimum k such that I(F) is definable in k-variable first-order logic. The recognition problem of I(F), known as Induced Subgraph Isomorphism (for the pattern graph F), is solvable in time O(n^{W[F]}). Motivated by this fact, we are interested in determining or estimating the value of W[F]. Using Olariu\u27s characterization of paw-free graphs, we show that I(K_3+e) is definable by a first-order sentence of quantifier depth 3, where K_3+e denotes the paw graph. This provides an example of a graph F with W[F] strictly less than the number of vertices in F. On the other hand, we prove that W[F]=4 for all F on 4 vertices except the paw graph and its complement. If F is a graph on t vertices, we prove a general lower bound W[F]>(1/2-o(1))t, where the function in the little-o notation approaches 0 as t increases. This bound holds true even for a related parameter W^*[F], which is defined as the minimum k such that I(F) is definable in the k-variable infinitary logic. We show that W^*[F] can be strictly less than W[F]. Specifically, W^*[P_4]=3 for P_4 being the path graph on 4 vertices