Homomorphisms between relational structures are not only fundamental mathematical objects, but are also of great importance in an applied computational context. Indeed, constraint satisfaction problems (CSPs), a wide class of algorithmic problems that occur in many different areas of computer science such as artificial intelligence or database theory, may be viewed as asking for homomorphisms between two relational structures [FedVar98]. In a logical setting, homomorphisms may be viewed as witnesses for positive primitive formulas in a relational language. As we shall see, homomorphisms, or more precisely the numbers of homomorphisms between two structures, are also related to a fundamental computational problem of statistical physics.
In this article, we are concerned with the complexity of counting homomorphisms from a given structure A to a fixed structure B. Actually, we are mainly interested in a generalization of this problem to weighted homomorphisms (or partition functions). We almost exclusively focus on graphs. The first part of the article is a short survey of what is known about the problem. In the second part, we give a proof of a theorem due to Bulatov and the first author of this paper [BulGro05], which classifies the complexity of partition functions described by matrices with non-negative entries. The proof we give here is essentially the same as the original one, with a few shortcuts due to [Thu09], but it is phrased in a different, more graph theoretical language that may make it more accessible to most readers.
A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H. Hell and Nesetril gave a complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a com…
Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular, independent sets and colourings. In this article we study the complexity of $\#_p\mathsf{H}$OMS$\mathsf{T}$O$H$, the problem of counting graph homomorphisms from an input graph to a graph $H$ modulo a prime number $p$. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the input graph. Many intractable cases in non-modular counting become tractabl…
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 study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, …
We study the problem of counting the number of homomorphisms from an input graph $G$ to a fixed (quantum) graph $\bar{H}$ in any finite field of prime order $\mathbb{Z}_p$. The subproblem with graph $H$ was introduced by Faben and Jerrum [ToC'15] and its complexity is subject to a growing series of research articles, e.g. the work of Focke, Goldberg, Roth, and Zivný [SIDMA'21] and the work of Bulatov and Kazeminia [STOC'22], subsequent to this article's conference version. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphi…
Many natural combinatorial quantities can be expressed by counting the number of homomorphisms to a fixed relational structure. For example, the number of 3-colorings of an undirected graph $G$ is equal to the number of homomorphisms from $G$ to the $3$-clique. In this setup, the structure receiving the homomorphisms is often referred to as a template; we use the term template function to refer to a function, from structures to natural numbers, that is definable as the number of homomorphisms to a fixed template. There is a literature that studies the complexity of template functions. The pres…
Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known abou…
Focke, Goldberg, and Živný (arXiv 2017) prove a complexity dichotomy for the problem of counting surjective homomorphisms from a large input graph G without loops to a fixed graph H that may have loops. In this note, we give a short proof of a weaker result: Namely, we only prove the #P-hardness of the more general problem in which G may have loops. Our proof is an application of a powerful framework of Lovász (2012), and it is analogous to proofs of Curticapean, Dell, and Marx (STOC 2017) who studied the "dual" problem in which the pattern graph G is small and the host graph H is the input. I…
We prove a complexity dichotomy theorem for counting planar graph homomorphisms of domain size 3. Given any 3 by 3 real valued symmetric matrix $H$ defining a graph homomorphism from all planar graphs $G \mapsto Z_H(G)$, we completely classify the computational complexity of this problem according to the matrix $H$. We show that for every $H$, the problem is either polynomial time computable or \#P-hard. The P-time computable cases consist of precisely those that are P-time computable for general graphs (a complete classification is known) or computable by Valiant's holographic algorithm via m…
The Parameterised Complexity of Temporal Motif Counting, and a Lovász-Style Isomorphism Theorem
Indian Institute of Technology Palakkad（印度理工学院帕拉卡德分校）；School of Computing Science, University of Glasgow（格拉斯哥大学计算科学学院）；School of Electronic Engineering and Computer Science, Queen Mary University of London（伦敦大学玛丽女王学院电子工程与计算机科学系）
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 consider the $\#\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\#\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree $2$ on one side. In our proofs, we use that $k$-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of $k$ length-$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edge-injective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edge-injective h…
Counting homomorphisms of a constant sized pattern graph $H$ in an input graph $G$ is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input $G$ and the pattern $H$. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of $H$,…
Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(H) --- the problem of finding the number of homomorphisms from a given graph G to the graph H. Not only the complexity of this basic problem is known, but also of its many variants for digraphs, more general relational structures, graphs with weights, and others. In this paper we consider a modification of #GraphHom(H), the #_p GraphHom(H) problem, p a prime …
In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there is no algorithm that solves this problem in time O(|V(H)|^{o(|V(G)|)}. This, however, does not rule out the possibility that faster algorithms exist for restricted problems of this kind. Wahlstrom proved that #GraphHom can be solved in plain exponential time, that is, in time k^{|V(G)|+V(H)|}\poly(|V(H)|,|V(G)|) provided H has clique width k. We generalize t…
Counting homomorphisms from a graph $H$ into another graph $G$ is a fundamental problem of (parameterized) counting complexity theory. In this work, we study the case where \emph{both} graphs $H$ and $G$ stem from given classes of graphs: $H\in \mathcal{H}$ and $G\in \mathcal{G}$. By this, we combine the structurally restricted version of this problem, with the language-restricted version. Our main result is a construction based on Kneser graphs that associates every problem $\tt P$ in $\#\mathsf{W[1]}$ with two classes of graphs $\mathcal{H}$ and $\mathcal{G}$ such that the problem $\tt P$ is…
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 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…
The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs $G$, $H$, and lists $L(v)\subseteq V(H)$ for every $v\in V(G)$, a {\em list homomorphism} is a function $f:V(G)\to V(H)$ that preserves the edges (i.e., $uv\in E(G)$ implies $f(u)f(v)\in E(H)$) and respects the lists (i.e., $f(v)\in L(v))$. Standard techniques show that if $G$ is given with a tree decomposition of width $t$, then the number of list homomorphisms can be counted in time $|V(H)|^t\cdot n^{\mathcal{O}…
We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in arXiv:1811.03126 [cs.CC]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In ano…
Given a symmetric matrix $M\in \{0,1,*\}^{D\times D}$, an $M$-partition of a graph $G$ is a function from $V(G)$ to $D$ such that no edge of $G$ is mapped to a $0$ of $M$ and no non-edge to a $1$. We give a computer-assisted proof that, when $|D|=4$, the problem of counting the $M$-partitions of an input graph is either in FP or is #P-complete. Tractability is proved by reduction to the related problem of counting list $M$-partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small nu…
Symmetric Parameterised Holants on Hypergraphs: Towards a Classification for Parameterised VCSPs
Hasso Plattner Institute, University of Potsdam（波茨坦大学哈索·普拉特纳研究所）；School of Electronic Engineering and Computer Science, Queen Mary University of London（伦敦大学玛丽女王学院电子工程与计算机科学学院）
We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $Γ$ and a degree bound $Δ$, we study the complexity of #CSP$_Δ(Γ)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $Γ$ and whose variables can appear at most $Δ$ times. Our main result shows that: (i) if every function in $Γ$ is affine, then #CSP$_Δ(Γ)$ is in FP for all $Δ$, (ii) otherwise, if every function in $Γ$ is in a class called IM$_2$, then for all sufficiently large $Δ$, #…
Minimum cost homomorphism problems can be viewed as a generalization of list homomorphism problems. They also extend two well-known graph colouring problems: the minimum colour sum problem and the optimum cost chromatic partition problem. In both of these problems, the cost function meets an additional constraint: the cost of using a specific colour is the same for every vertex of the input graph. We study minimum cost homomorphism problems with cost functions constrained to have this property. Clearly, when the standard minimum cost homomorphism problem is polynomial, then the problem with co…
Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $π_1(S^3\setminus K) \to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping class…
