Counting Graph Homomorphisms Research Articles

Complexity Classification of Counting Graph Homomorphisms Modulo a Prime Number

Complexity Classification of Counting Graph Homomorphisms Modulo a Prime Number

The computational complexity of Holant problems on 3-regular graphs

Holant problem is a framework to study counting problems, which is expressive enough to contain Counting Graph Homomorphisms (#GH) and Counting Constraint Satisfaction Problems (#CSP) as special cases. In the present paper, we classify the computational complexity of Holant problems on 3-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we prove that Holant problem on 3-regular graphs is #P-hard except for the signature is not genuinely entangled, A-transformable, P-transformable or vanishing, in which cases the problem is tractable.

Parameterised Counting in Logspace

Logarithmic space-bounded complexity classes such as textbf{L} and textbf{NL} play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators textbf{para}_{textbf{W}} and textbf{para}_beta for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators textbf{para}_{textbf{W}} and textbf{para}_beta by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, textbf{para}_{{textbf{W}}[1]} and textbf{para}_{beta {textbf{tail}}}. Then, we consider counting versions of all four operators and apply them to the class textbf{L} . We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is #textbf{para}_{beta {textbf{tail}}}textbf{L} -hard and can be written as the difference of two functions in #textbf{para}_{beta {textbf{tail}}}textbf{L} . These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #textbf{para}_{beta {textbf{tail}}}textbf{L} under parameterised logspace parsimonious reductions coincides with #textbf{para}_beta textbf{L} . In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.

Counting Homomorphisms to Trees Modulo a Prime

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 H OMS 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 target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree H and every prime p the problem # p H OMS T O H is either polynomial time computable or # p P-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of # p H OMS T O H are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p . These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p .

Parameterized Counting of Partially Injective Homomorphisms

We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and #mathsf {W[1]}-complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.

Complexity of Counting CSP with Complex Weights

We give a complexity dichotomy theorem for the counting constraint satisfaction problem (#CSP in short) with algebraic complex weights. To this end, we give three conditions for its tractability. Let F be any finite set of algebraic complex-valued functions defined on an arbitrary finite domain. We show that #CSP( F ) is solvable in polynomial time if all three conditions are satisfied and is #P-hard otherwise. Our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination: (a) the problem of counting graph homomorphisms is the special case when F has a single symmetric binary function [Dyer and Greenhill 2000; Bulatov and Grohe 2005; Goldberg et al. 2010; Cai et al. 2013]; (b) the problem of counting directed graph homomorphisms is the special case when F has a single but not necessarily symmetric binary function [Dyer et al. 2007; Cai and Chen 2010]; (c) the unweighted form of #CSP is when all functions in F take values in {0, 1} [Bulatov 2008; Dyer and Richerby 2013].

Matroid invariants and counting graph homomorphisms

The number of homomorphisms from a finite graph F to the complete graph Kn is the evaluation of the chromatic polynomial of F at n. Suitably scaled, this is the Tutte polynomial evaluation T(F;1−n,0) and an invariant of the cycle matroid of F. De la Harpe and Jaeger [8] asked more generally when is it the case that a graph parameter obtained from counting homomorphisms from F to a fixed graph G depends only on the cycle matroid of F. They showed that this is true when G has a generously transitive automorphism group (examples include Cayley graphs on an abelian group, and Kneser graphs).Using tools from multilinear algebra, we prove the converse statement, thus characterizing finite graphs G for which counting homomorphisms to G yields a matroid invariant. We also extend this result to finite weighted graphs G (where to count homomorphisms from F to G includes such problems as counting nowhere-zero flows of F and evaluating the partition function of an interaction model on F).

Spin systems on [formula omitted]-regular graphs with complex edge functions

Let k≥1 be an integer and let h=[h(0,0)h(0,1)h(1,0)h(1,1)] be a complex-valued symmetric function on domain {0,1} (i.e., where h(0,1)=h(1,0)). We introduce a new technique, called a syzygy, and prove a dichotomy theorem for the following class of problems, specified by k and h: given an arbitrary k-regular graph G=(V,E), where the function h is attached to each edge, compute Z(G)=∑σ:V→{0,1}∏{u,v}∈Eh(σ(u),σ(v)). Z(⋅) is known as the partition function of the spin system, also known as counting graph homomorphisms on domain size two, and is a special case of Holant problems. The dichotomy theorem gives a complete classification of the computational complexity of this problem, depending on k and h. The dependence on k and h is explicit.

Counting Subgraphs via Homomorphisms

We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well-known results in algorithms and combinatorics, including the recent algorithm of Björklund, Husfeldt, and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn et al. for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color-coding-based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions, and the color coding technique, we obtain the following new results. The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time $ 2^{n+o(n)} $, in particular, in time $\mathcal{O}(2^{n}n^3)$ for trees and in time $2^{n+\mathcal{O}(\sqrt{n})}$ for planar graphs. Counting all maximum planar subgraphs, subgraphs of bounded genus, or more generally subgraphs excluding a fixed graph M as a minor can be done in $2^{\mathcal{O}(n)}$ time. Counting all subtrees with a given maximum degree (a generalization of counting Hamiltonian paths) of a given graph can be done in time $2^{\mathcal{O}(n)}$. A generalization of Ryser's formula is, Let G be a graph with an independent set of size $\ell$. Then the number of perfect matchings in G can be found in time $\mathcal{O}(2^{n-\ell} n^3)$. Let ${\cal H}$ be a graph class excluding a fixed graph M as a minor. Then the maximum number of vertex disjoint subgraphs from ${\cal H}$ in a graph G on n vertices can be found in time $2^{\mathcal{O}(n)}$. In order to show this, we prove that there exists a constant $c_M$ depending only on M such that the number of nonisomorphic n-vertex graphs in ${\cal H}$ is at most $c_M^n$. Let F be a k-vertex graph of treewidth t and let G be an n-vertex graph. A subgraph of G isomorphic to F (if one exists) can be found in $\mathcal{O}(4.32^k \cdot k \cdot t \cdot n^{t+1})$ expected time using $\mathcal{O}(\log{k} \cdot n^{t+1})$ space.

Guest column

The complexity of counting problems has been studied intensively over the years. During the past decade, significant progress has taken place and a series of successively more powerful complexity dichotomies have been discovered. Given a class of counting problems, a dichotomy states that every problem in the class is either in polynomial time or #P-hard. In this guest column, we briefly survey two threads of recent developments in counting graph homomorphisms and the counting constraint satisfaction problem.

A Complexity Dichotomy For Hypergraph Partition Functions

We consider the complexity of counting homomorphisms from an r-uniform hypergraph G to a symmetric r-ary relation H. We give a dichotomy theorem for r > 2, showing for which H this problem is in FP and for which H it is #P-complete. This generalizes a theorem of Dyer and Greenhill (2000) for the case r = 2, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation H is weighted, and the goal is to compute the partition function, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of r sites interact symmetrically. In the weighted case, our dichotomy theorem generalizes a result of Bulatov and Grohe (2005) for graphs, where r = 2. When r = 2, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all r > 2 the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.

Towards a dichotomy theorem for the counting constraint satisfaction problem

The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal’tsev polymorphism, that is a ternary operation m(x,y,z) such that m(x,y,y)=m(y,y,x)=x. This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.

Corrigendum: The complexity of counting graph homomorphisms

AbstractWe close a gap in the proof of Theorem 4.1 in our paper “The complexity of counting graph homomorphisms” [Random Structures Algorithms 17 (2000), 260–289]. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

The complexity of counting graph homomorphisms

The complexity of counting graph homomorphisms

The complexity of counting graph homomorphisms

The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalization of graph coloring, with important applications in statistical physics. The problem of deciding whether any homomorphism exists was considered by Hell and Nesetřil. They showed that decision is NP-complete unless H has a loop or is bipartite; otherwise it is in P. We consider the problem of exactly counting such homomorphisms and give a similarly complete characterization. We show that counting is #P-complete unless every connected component of H is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite graph; otherwise it is in P. We prove further that this remains true when G has bounded degree. In particular, our theorems provide the first proof of #P-completeness of the partition function of certain models from statistical physics, such as the Widom–Rowlinson model, even in graphs of maximum degree 3. Our results are proved using a mixture of spectral analysis, interpolation, and combinatorial arguments. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 260–289, 2000