We present an algorithm to count the number of occurrences of a pattern graph $H$ as an induced subgraph in a host graph $G$. If $G$ belongs to a bounded expansion class, the algorithm runs in linear time. Our design choices are motivated by the need for an approach that can be engineered into a practical implementation for sparse host graphs.
Specifically, we introduce a decomposition of the pattern $H$ called a counting dag $\vec C(H)$ which encodes an order-aware, inclusion-exclusion counting method for $H$. Given such a counting dag and a suitable linear ordering $\mathbb G$ of $G$ as input, our algorithm can count the number of times $H$ appears as an induced subgraph in $G$ in time $O(\|\vec C\| \cdot h \text{wcol}_{h}(\mathbb G)^{h-1} |G|)$, where $\text{wcol}_h(\mathbb G)$ denotes the maximum size of the weakly $h$-reachable sets in $\mathbb G$. This implies, combined with previous results, an algorithm with running time $O(4^{h^2}h (\text{wcol}_h(G)+1)^{h^3} |G|)$ which only takes $H$ and $G$ as input.
We note that with a small modification, our algorithm can instead use strongly $h$-reachable sets with running time $O(\|\vec C\| \cdot h \text{col}_{h}(\mathbb G)^{h-1} |G|)$, resulting in an overall complexity of $O(4^{h^2}h \text{col}_h(G)^{h^2} |G|)$ when only given $H$ and $G$.
Because orderings with small weakly/strongly reachable sets can be computed relatively efficiently in practice [11], our algorithm provides a promising alternative to algorithms using the traditional $p$-treedepth colouring framework [13]. We describe preliminary experimental results from an initial open source implementation which highlight its potential.
Computing subgraph frequencies is a fundamental task that lies at the core of several network analysis methodologies, such as network motifs and graphlet-based metrics, which have been widely used to categorize and compare networks from multiple domains. Counting subgraphs is however computationally very expensive and there has been a large body of work on efficient algorithms and strategies to make subgraph counting feasible for larger subgraphs and networks. This survey aims precisely to provide a comprehensive overview of the existing methods for subgraph counting. Our main contribution is …
In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the …
We consider the problem of counting all $k$-vertex subgraphs in an input graph, for any constant $k$. This problem (denoted sub-cnt$_k$) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for sub-cnt$_k$. Towards a better understanding o…
We give an algorithm for counting self-avoiding walks or self-avoiding polygons that runs in time $\exp(C\sqrt{n\log n})$ on 2-dimensional lattices and time $\exp(C_dn^{(d-1)/d}\log n)$ on $d$-dimensional lattices for $d>2$.
We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an $n^{f(t,s)}$-time algorithm to compute modulo $2^t$ the number of subgraph occurrences of patterns that are $s$ vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo $2^t$. Complementing our algorit…
This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamic graph. Specifically, our algorithms maintain the number of paths of length three in deterministic amortized $\mathcal{O}(m^\frac{1}{2})$ update time, and any other connected four-vertex subgraph which is not a clique in deterministic amortized update time $\mathcal{O}(m^\frac{2}{3})$. Queries can be answered in constant time. We also study the query times for subgraphs containing an arbitrary edge that is supplied only with the query as well as the case where o…
Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot…
This paper initializes the study of {\em range subgraph counting} and {\em range subgraph listing}, both of which are motivated by the significant demands in practice to perform graph analytics on subgraphs pertinent to only selected, as opposed to all, vertices. In the first problem, there is an undirected graph $G$ where each vertex carries a real-valued attribute. Given an interval $q$ and a pattern $Q$, a query counts the number of occurrences of $Q$ in the subgraph of $G$ induced by the vertices whose attributes fall in $q$. The second problem has the same setup except that a query needs …
Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph $G=(V, E)$ with $n$ vertices, $m$ edges and $T$…
Estimating the number of subgraphs in data streams is a fundamental problem that has received great attention in the past decade. In this paper, we give improved streaming algorithms for approximately counting the number of occurrences of an arbitrary subgraph $H$, denoted $\# H$, when the input graph $G$ is represented as a stream of $m$ edges. To obtain our algorithms, we provide a generic transformation that converts constant-round sublinear-time graph algorithms in the query access model to constant-pass sublinear-space graph streaming algorithms. Using this transformation, we obtain the f…
Subgraph counting is a fundamental task in network analysis. Typically, algorithmic work is on total counting, where we wish to count the total frequency of a (small) pattern subgraph in a large input data set. But many applications require local counts (also called vertex orbit counts) wherein, for every vertex $v$ of the input graph, one needs the count of the pattern subgraph involving $v$. This provides a rich set of vertex features that can be used in machine learning tasks, especially classification and clustering. But getting local counts is extremely challenging. Even the easier proble…
We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices…
Faster Algorithms for Finding and Counting Subgraphs
In this paper we study a natural generalization of both { \sc $k$ -Path} and { \sc $k$ -Tree} problems, namely, the { \sc Subgraph Isomorphism} problem. In the { \sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$ . We show that if the treewidth of $F$ is at most $t$ , then there is a randomized algorithm for the { \sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$ . To do so, we associate a new multivariate {Homomorphism polynomial…
The $h$-index is a metric used to measure the impact of a user in a publication setting, such as a member of a social network with many highly liked posts or a researcher in an academic domain with many highly cited publications. Specifically, the $h$-index of a user is the largest integer $h$ such that at least $h$ publications of the user have at least $h$ units of positive feedback. We design an algorithm that, given query access to the $n$ publications of a user and each publication's corresponding positive feedback number, outputs a $(1\pm \varepsilon)$-approximation of the $h$-index of t…
We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a $k$-vertex graph $H$ in a $d$-degenerate $n$-vertex graph $G$. Our main result establishes exhaustive and explicit complexity classifications for counting subgraphs and induced subgraphs. We show that the (not necessarily induced) copies of $H$ in $G$ can be counted in time $f(k,d)\cdot n^{\max(\mathsf{imn}(H),1)}\cdot \log n$, where $f$ is some computable function and $\mathsf{imn}(H)$ is the size of the largest induced matching of $H$. Whenever the class of allowed patterns has unbo…
We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that $H$ is easy if there is a linear-time algorithm for counting the number of copies of $H$ in an inp…
A $k$-motif (or graphlet) is a subgraph on $k$ nodes in a graph or network. Counting of motifs in complex networks has been a well-studied problem in network analysis of various real-word graphs arising from the study of social networks and bioinformatics. In particular, the triangle counting problem has received much attention due to its significance in understanding the behavior of social networks. Similarly, subgraphs with more than 3 nodes have received much attention recently. While there have been successful methods developed on this problem, most of the existing algorithms are not scala…
Given a graph property $Φ$, we consider the problem $\mathtt{EdgeSub}(Φ)$, where the input is a pair of a graph $G$ and a positive integer $k$, and the task is to decide whether $G$ contains a $k$-edge subgraph that satisfies $Φ$. Specifically, we study the parameterized complexity of $\mathtt{EdgeSub}(Φ)$ and of its counting problem $\#\mathtt{EdgeSub}(Φ)$ with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties $Φ$: the decision problem $\mathtt{EdgeSub}(Φ)$ always admits an FPT algorithm and the counting problem $\#\mathtt{EdgeSub}(Φ)$ al…
The problem of counting small subgraphs, and specifically cycles, in the streaming model received a lot of attention over the past few years. In this paper, we consider arbitrary order insertion-only streams, improving over the state-of-the-art result on counting 4-cycles. Our algorithm computes a $(1+ε)$-approximation by taking three passes over the stream and using space $O(\frac{m \log n}{ε^2 T^{1/3}})$, where $m$ is the number of edges in the graph and $T$ is the number of 4-cycles.
Counting instances of specific subgraphs in a larger graph is an important problem in graph mining. Finding cliques of size k (k-cliques) is one example of this NP-hard problem. Different algorithms for clique counting avoid counting the same clique multiple times by pivoting or ordering the graph. Ordering-based algorithms include an ordering step to direct the edges in the input graph, and a counting step, which is dominated by building node or edge-induced subgraphs. Of the ordering-based algorithms, kClist is the state-of-the art algorithm designed to work on sparse real-world graphs. Desp…
We present a new parallel algorithm for $k$-clique counting/listing that has polylogarithmic span (parallel time) and is work-efficient (matches the work of the best sequential algorithm) for sparse graphs. Our algorithm is based on computing low out-degree orientations, which we present new linear-work and polylogarithmic-span algorithms for computing in parallel. We also present new parallel algorithms for producing unbiased estimations of clique counts using graph sparsification. Finally, we design two new parallel work-efficient algorithms for approximating the $k$-clique densest subgraph,…
We study the problem of counting all cycles or self-avoiding walks (SAWs) on triangulated planar graphs. We present a subexponential $2^{O(\sqrt{n})}$ time algorithm for this counting problem. Among the technical ingredients used in this algorithm are the planar separator theorem and a delicate analysis using pairs of Motzkin paths and Motzkin numbers. We can then adapt this algorithm to uniformly sample SAWs, in subexponential time. Our work is motivated by the problem of gerrymandered districting maps.
Inductive $k$-independent graphs generalize chordal graphs and have recently been advocated in the context of interference-avoiding wireless communication scheduling. The NP-hard problem of finding maximum-weight induced $c$-colorable subgraphs, which is a generalization of finding maximum independent sets, naturally occurs when selecting $c$ sets of pairwise non-conflicting jobs (modeled as graph vertices). We investigate the parameterized complexity of this problem on inductive $k$-independent graphs. We show that the Independent Set problem is W[1]-hard even on 2-simplicial 3-minoes---a sub…
