For a fixed "pattern" graph $G$, the $\textit{colored $G$-subgraph isomorphism problem}$ (denoted $\mathrm{SUB}(G)$) asks, given an $n$-vertex graph $H$ and a coloring $V(H) \to V(G)$, whether $H$ contains a properly colored copy of $G$. The complexity of this problem is tied to parameterized versions of $\mathit{P}$ ${=}?$ $\mathit{NP}$ and $\mathit{L}$ ${=}?$ $\mathit{NL}$, among other questions. An overarching goal is to understand the complexity of $\mathrm{SUB}(G)$, under different computational models, in terms of natural invariants of the pattern graph $G$.
In this paper, we establish a close relationship between the $\textit{formula complexity}$ of $\mathrm{SUB}$ and an invariant known as $\textit{tree-depth}$ (denoted $\mathrm{td}(G)$). $\mathrm{SUB}(G)$ is known to be solvable by monotone $\mathit{AC^0}$ formulas of size $O(n^{\mathrm{td}(G)})$. Our main result is an $n^{\tildeΩ(\mathrm{td}(G)^{1/3})}$ lower bound for formulas that are monotone $\textit{or}$ have sub-logarithmic depth. This complements a lower bound of Li, Razborov and Rossman (SICOMP 2017) relating tree-width and $\mathit{AC^0}$ circuit size. As a corollary, it implies a stronger homomorphism preservation theorem for first-order logic on finite structures (Rossman, ITCS 2017).
The technical core of this result is an $n^{Ω(k)}$ lower bound in the special case where $G$ is a complete binary tree of height $k$, which we establish using the $\textit{pathset framework}$ introduced in (Rossman, SICOMP 2018). (The lower bound for general patterns follows via a recent excluded-minor characterization of tree-depth (Czerwiński et al, arXiv:1904.13077).) Additional results of this paper extend the pathset framework and improve upon both, the best known upper and lower bounds on the average-case formula size of $\mathrm{SUB}(G)$ when $G$ is a path.
We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …
Let $v(F)$ denote the number of vertices in a fixed connected pattern graph $F$. We show an infinite family of patterns $F$ such that the existence of a subgraph isomorphic to $F$ is expressible by a first-order sentence of quantifier depth $\frac23\,v(F)+1$, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any $F$ with using less than $\frac23\,v(F)-2$ first-order variables.
We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P(…
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'95], 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 $O(n^{tw(H)+1-\varepsilon})$ or even faster (e.g. for $k$-cliques), it is not known whether such improvements are…
Let $F$ be a connected graph with $\ell$ vertices. The existence of a subgraph isomorphic to $F$ can be defined in first-order logic with quantifier depth no better than $\ell$, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs $K_\ell$ and $K_{\ell-1}$. We show that, for some $F$, the existence of an $F$ subgraph in \emph{sufficiently large} connected graphs is definable with quantifier depth $\ell-3$. On the other hand, this is never possible with quantifier depth better than $\ell/2$. If we, however, consider definitions over conne…
Let $H$ be a fixed graph on $n$ vertices. Let $f_H(G) = 1$ iff the input graph $G$ on $n$ vertices contains $H$ as a (not necessarily induced) subgraph. Let $α_H$ denote the cardinality of a maximum independent set of $H$. In this paper we show: \[Q(f_H) = Ω\left(\sqrt{α_H \cdot n}\right),\] where $Q(f_H)$ denotes the quantum query complexity of $f_H$. As a consequence we obtain a lower bounds for $Q(f_H)$ in terms of several other parameters of $H$ such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that $Q(f_H)…
We study the time complexity of induced subgraph isomorphism problems where the pattern graph is fixed. The earliest known example of an improvement over trivial algorithms is by Itai and Rodeh (1978) who sped up triangle detection in graphs using fast matrix multiplication. This algorithm was generalized by Nešetřil and Poljak (1985) to speed up detection of k-cliques. Improved algorithms are known for certain small-sized patterns. For example, a linear-time algorithm is known for detecting length-4 paths. In this paper, we give the first pattern detection algorithm that improves upon Nešetři…
For any fixed graph $G$, the subgraph isomorphism problem asks whether an $n$-vertex input graph has a subgraph isomorphic to $G$. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted $G$-$\mathsf{SUB}$, and then solves $G$-$\mathsf{SUB}$ in time $O(n^{tw(G)+1})$ where $tw(G)$ is the treewidth of $G$. Marx (2010) conjectured that $G$-$\mathsf{SUB}$ requires time $Ω(n^{\mathrm{const}\cdot tw(G)})$ and, assuming the Exponential Time Hypothesis, proved a lower bound of $Ω(n^{\mathrm{const}\cdot emb(G)})$ for a certain gr…
Given two graphs $H$ and $G$, the Subgraph Isomorphism problem asks if $H$ is isomorphic to a subgraph of $G$. While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time $2^{O(|V(H)|)}\cdot n^{O(\tw(H))}$ using the color-coding technique of Alon, Yuster, and Zwick; (2) in time $f(|V(H)|,\tw(G))\cdot n$ using Courcelle's Theorem; (3) in time $f(|V(H)|,\genus(G))\cdot n$ using a result on first-order model checking by Frick and Grohe; or (4) in time $f(\maxdeg(H))\cdot n^{O(\tw(G)})$ for connected $H$ using the…
An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as $H$-graphs, and, in particular, every graph is an $H$-graph for a suitable graph $H$. An $H$-graph is called proper if it has a representation where no subgraph properly contains another. We consider the recognition and isomorphism problems for proper $U$-graphs where $U$ is a unicylic graph. We prove that testing whether a graph is a (proper) $U$-graph, for some $U$…
The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a fruitful approach to the Graph Isomorphism problem. 2-WL corresponds to the original algorithm suggested by Weisfeiler and Leman over 50 years ago. 1-WL is the classical color refinement routine. Indistinguishability by $k$-WL is an equivalence relation on graphs that is of fundamental importance for isomorphism testing, descriptive complexity theory, and graph similarity testing which is also of some relevance in artificial intelligence. Focusing on dimensions $k=1,2$, we investigate subgraph patterns whose counts are $k$-WL invaria…
In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these cannot be solved in time $2^{o(|s|/\log{|s|})}$, unless the Exponential Time Hypothesis fails. These results are used to obtain simplified hardness results for several graph embedding problems, on more restricted graph classes than previously known: assuming the Exponential Time Hypothesis, there do not exist algorithms that run in $2^{o(n/\log n)}$ time for Subg…
On the Expressibility of the Reconstructional Color Refinement
The Institute of Mathematical Sciences (HBNI)（数学科学研究所（HBNI））；Chennai Mathematical Institute（金奈数学研究所）；Institut für Informatik, Humboldt-Universität zu Berlin（柏林洪堡大学计算机科学研究所）；IAPMM
In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\GI{}) for several parameterizations. Let $\mathcal{H}=\{H_1,H_2,\cdots,H_l\}$ be a finite set of graphs where $|V(H_i)|\leq d$ for all $i$ and for some constant $d$. Let $\mathcal{G}$ be an $\mathcal{H}$-free graph class i.e., none of the graphs $G\in \mathcal{G}$ contain any $H \in \mathcal{H}$ as an induced subgraph. We show that \GI{} parameterized by vertex deletion distance to $\mathcal{G}$ is in a parameterized version of $\AC^1$, denoted $\PL$-$\AC^1$, provided the colored graph isomorphism …
We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most $11\log_2 n+43$. As a consequence, a canonic form of such graphs is computable in $AC^1$ by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in $AC^1$.
The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special c…
The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on…
