Given a graph property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $Φ$. The search for explicit criteria on $Φ$ ensuring that $\#\mathsf{IndSub}(Φ)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfler et al. [MFCS 19], not much is known.
In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result.
Given a $k$-node pattern graph $H$ and an $n$-node host graph $G$, the subgraph counting problem asks to compute the number of copies of $H$ in $G$. In this work we address the following question: can we count the copies of $H$ faster if $G$ is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs. This decomposition gives a dynamic program for counting the homomorphisms of $H$ in $G$ by exploiting the degeneracy of $G$, which allows us to beat the state-of-the-art subgra…
A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting …
Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area. Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of these subgraphs in some sense; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting problem. The p…
We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that the problems of counting even and odd k-vertex induced subgr…
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…
We introduce a class of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focusses on the case in which Φdefines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φdescribes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that the…
We investigate the problem $\#\mathsf{IndSub}(Φ)$ of counting all induced subgraphs of size $k$ in a graph $G$ that satisfy a given property $Φ$. This continues the work of Jerrum and Meeks who proved the problem to be $\#\mathrm{W[1]}$-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ is hard for $\#\mathrm{W[…
We study the computational complexity of the problem $\#\text{IndSub}(Φ)$ of counting $k$-vertex induced subgraphs of a graph $G$ that satisfy a graph property $Φ$. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): - If a hereditary property $Φ$ is true for all graphs, or if it is true only for finitely many graphs, then $\#\text{IndSub}(Φ)$ is solvable in polynomial time. - Otherwise, $\#\text{IndSub}(Φ)$ is $\#\mathsf{W[1]}$-complete when parameterised by $…
For a class $\mathcal{H}$ of graphs, #Sub$(\mathcal{H})$ is the counting problem that, given a graph $H\in \mathcal{H}$ and an arbitrary graph $G$, asks for the number of subgraphs of $G$ isomorphic to $H$. It is known that if $\mathcal{H}$ has bounded vertex-cover number (equivalently, the size of the maximum matching in $\mathcal{H}$ is bounded), then #Sub$(\mathcal{H})$ is polynomial-time solvable. We complement this result with a corresponding lower bound: if $\mathcal{H}$ is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub$(\mathcal{H})$ is #W[1]-ha…
We present a framework for the complexity classification of parameterized counting problems that can be formulated as the summation over the numbers of homomorphisms from small pattern graphs H_1,...,H_l to a big host graph G with the restriction that the coefficients correspond to evaluations of the Möbius function over the lattice of a graphic matroid. This generalizes the idea of Curticapean, Dell and Marx [STOC 17] who used a result of Lovász stating that the number of subgraph embeddings from a graph H to a graph G can be expressed as such a sum over the lattice of partitions of H. In the…
We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs $H$, the number of $H$-copies (induced or not) in an input graph $G$, and the number of homomorphisms from $H$ to $G$. Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs $H$ in host graphs $G$: For graphs $H$ on $k$ edges, we show how to count subgraph copies of $H$ in time $k^{O(k)}\c…
We consider parameterised subgraph-counting problems of the following form: given a graph G, how many k-tuples of its vertices have a given property? A number of such problems are known to be #W[1]-complete; here we substantially generalise some of these existing results by proving hardness for two large families of such problems. We demonstrate that it is #W[1]-hard to count the number of k-vertex subgraphs having any property where the number of distinct edge-densities of labelled subgraphs that satisfy the property is o(k^2). In the special case that the property in question depends only on…
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…
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…
Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes as input a single string $x \in \{0,1\}^n$ and outputs $1$ if and only if $x_1 + \cdots + x_n \geq t$. We refer to circuits that are composed of THR gates as monotone majority circuits. The main result of this paper is an exponential lower bound on the size of b…
We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph m…
We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete.
We study the problem $\#\mathrm{EdgeSub}(Φ)$ of counting $k$-edge subgraphs satisfying a given graph property $Φ$ in a large host graph $G$. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of $p$-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coeffi…
A retraction is a homomorphism from a graph $G$ to an induced subgraph $H$ of $G$ that is the identity on $H$. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length $4$). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class $\#\mathrm{BIS}$. As retractions generalise h…
We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$, "forbidden subgraph $H$" is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-ε}$ edges are eventually evasive. ($n$ is the number of vertices.) While Chowla's conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's $L$ functions), we show (b) with the bound $O(n…
