Homomorphism Problem Research Articles

The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

The generic homomorphism problem, which asks whether an input graph \(G\) admits a homomorphism into a fixed target graph \(H\) , has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of \(G\) (denoted \({\operatorname{cw}}\) ) for virtually all choices of \(H\) under the Strong Exponential Time Hypothesis. In particular, we identify a property of \(H\) called the signature number \(s(H)\) and show that for each \(H\) , the homomorphism problem can be solved in time \(\mathcal{O^{*}}(s(H)^{{\operatorname{cw}}})\) . Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each \(H\) that is either a projective core or a graph admitting a factorization with additional properties—allowing us to cover all possible target graphs under long-standing conjectures.

List homomorphisms to separable signed graphs

The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees [4] and reflexive signed graphs [25]. Irreflexive signed graphs are in a certain sense the heart of the problem, as noted by a recent paper of Kim and Siggers. We focus on a special class of irreflexive signed graphs, namely those in which the unicoloured edges form a spanning path or cycle, which we call separable signed graphs. We classify the complexity of list homomorphisms to these separable signed graphs; we believe that these signed graphs will play an important role for the general resolution of the irreflexive case. We also relate our results to a conjecture of Kim and Siggers concerning the special case of semi-balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

Pliability and Approximating Max-CSPs

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

Bounded Degree Nonnegative Counting CSP

Constraint satisfaction problems (CSP) encompass an enormous variety of computational problems. In particular, all partition functions from statistical physics, such as spin systems, are special cases of counting CSP (#CSP). We prove a complete complexity classification for every counting problem in #CSP with nonnegative valued constraint functions that is valid when every variable occurs a bounded number of times in all constraints. We show that, depending on the set of constraint functions \(\mathcal {F} \) , every problem in the complexity class # \(\text{CSP}(\mathcal {F}) \) defined by \(\mathcal {F} \) is either polynomial-time computable for all instances without the bounded occurrence restriction, or is #P-hard even when restricted to bounded degree input instances. The constant bound in the degree depends on \(\mathcal {F} \) . The dichotomy criterion on \(\mathcal {F} \) is decidable. As a second contribution, we prove a slightly modified but more streamlined decision procedure (from [14]) to test for the tractability of # \(\text{CSP}(\mathcal {F}) \) . This procedure on an input \(\mathcal {F} \) tells us which case holds in the dichotomy for # \(\text{CSP}(\mathcal {F}) \) . This more streamlined decision procedure enables us to fully classify a family of directed weighted graph homomorphism problems. This family contains both P-time tractable problems and #P-hard problems. To our best knowledge, this is the first family of such problems explicitly classified that are not acyclic , thereby the Lovász-goodness criterion of Dyer-Goldberg-Paterson [24] cannot be applied.

The Complexity of the Matroid Homomorphism Problem

We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.

Topology and Adjunction in Promise Constraint Satisfaction

The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a $c$-colouring of a graph that is promised to be $k$-colourable, where $c\geq k$. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.

List homomorphism problems for signed trees

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph (G,σ), equipped with lists L(v)⊆V(H),v∈V(G), of allowed images, to a fixed target signed graph (H,π). The complexity of the similar homomorphism problem without lists (corresponding to all lists being L(v)=V(H)) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when H is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and in a future companion paper we will illustrate this by using them to classify the complexity for certain irreflexive signed graphs. The structure of the signed trees in the polynomial cases is interesting, suggesting that the class of general signed graphs for which the problems are polynomial may have nice structure, analogous to the so-called bi-arc graphs (which characterised the polynomial cases of list homomorphisms to unsigned graphs).

Approximation approaches for rough hypersoft sets based on hesitant bipolar-valued fuzzy hypersoft relations on semigroups

In the hybrid context of hesitant bipolar-valued fuzzy hypersoft relations, the modern notion of extended roughness is constructed to rough approximations of hypersoft sets and fuzzy sets based on such context in this research. Then, corresponding examples are proposed, and further verified in connections between the hesitant bipolar-valued fuzzy hypersoft relations and the upper (resp., lower) rough approximations of hypersoft sets and fuzzy sets. Specifically, relationships are shown between the non-rough hypersoft sets (resp., non-rough fuzzy sets) and hesitant bipolar-valued fuzzy hypersoft reflexive relations together with hesitant bipolar-valued fuzzy hypersoft antisymmetric relations. To find the optimal multi-parameter of a hypersoft set such that the best choice exists, the notion of the set-valued measurement issues and decision-making algorithm for such objective is developed in the terms of rough set theory. Associated with the aforementioned accomplishments, the notion of novel models has been used to semigroups. Subsequently, the argumentation within relationships concerning the upper (resp., lower) rough approximations of hypersoft quasi-ideals and fuzzy quasi-ideals are proved under hypersoft homomorphism problems.

Graph Modification for Edge-Coloured and Signed Graph Homomorphism Problems: Parameterized and Classical Complexity

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from edge-coloured graph $G$ to edge-coloured graph $H$ is a vertex-mapping from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph $H$. The question we are interested in is: given an edge-coloured graph $G$, can we perform $k$ graph operations so that the resulting graph admits a homomorphism to $H$? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs $+$ and $-$. We denote the corresponding problems (parameterized by $k$) by VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. These problems generalise $H$-COLOURING (to decide if an input graph admits a homomorphism to a fixed target $H$). Our main focus is when $H$ is an edge-coloured graph with at most two vertices, a case that is already interesting as it includes problems such as VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph $H$, we give a P/NP-c complexity dichotomy for VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. We then address their parameterized complexity. We show that VD-$H$-COLOURING and ED-$H$-COLOURING for all such $H$ are FPT. In contrast, already for some $H$ of order 3, unless P=NP, none of the three problems is in XP, since 3-COLOURING is NP-c. We show that SW-$H$-COLOURING is different: there are three 2-edge-coloured graphs $H$ of order 2 for which SW-$H$-COLOURING is W-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$. For the other cases, SW-$H$-COLOURING is FPT.

Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets -- CS. $\circ$ Every partially ordered set has a cofinal well-founded subset -- CWF. $\circ$ Dilworth's decomposition theorem for infinite partially ordered sets of finite width -- DT. We also study a graph homomorphism problem and a problem due to A. Hajnal without AC. Further, we study a few statements restricted to linearly-ordered structures without AC.

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 .

Coloring invariants of knots and links are often intractable

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 $\pi_1(S^3\setminus K) \to G$ 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 groups actions on closed, unmarked surfaces, we now use braid group actions.

Algebra of Modular Systems: Containment and Equivalence

The Algebra of Modular System is a KR formalism that allows for combinations of modules written in multiple languages. Informally, a module represents a piece of knowledge. It can be given by a knowledge base, be an agent, an ASP, ILP, CP program, etc. Formally, a module is a class of structures over the same vocabulary. Modules are combined declaratively, using, essentially, operations of Codd's relational algebra. In this paper, we address the problem of checking modular system containment, which we relate to a homomorphism problem. We prove that, for a large class of modular systems, the containment problem (and thus equivalence) is in the complexity class NP, and therefore is solvable by, e.g., SAT solvers. We discuss conditions under which the problem is polynomial time solvable.

Coloring Problems on Bipartite Graphs of Small Diameter

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving $\textsf{NP}$-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$.
 Some of these results are obtained $\textsc{through}$ a proof that the Surjective $C_6$-Homomorphism problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is $\textsf{NP}$-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here.
 Finally, we prove that the $3$-Fall Coloring problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four, and prove that $\textsf{NP}$-completeness for diameter three would also imply $\textsf{NP}$-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].

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.

Fine-Grained Complexity of the Graph Homomorphism Problem for Bounded-Treewidth Graphs

For a fixed graph $H$, by Hom($H$) we denote the computational problem which asks whether a given graph $G$ admits a homomorphism to $H$, i.e., an edge-preserving mapping from $V(G)$ to $V(H)$. As ...

Homomorphisms into Loop-Threshold Graphs

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and edges into small image graphs. For the image graph $H_\text{ind}$, the graph on two adjacent vertices, one of which is looped, each homomorphism from $G$ to $H_\text{ind}$ corresponds to an independent set in $G$. It follows from the Kruskal-Katona theorem that the number of homomorphisms to $H_\text{ind}$ is maximized by the lex graph, whose edges form an initial segment of the lex order. 
 A loop-threshold graph is a graph built recursively from a single vertex, which may be looped or unlooped, by successively adding either a looped dominating vertex or an unlooped isolated vertex at each stage. Thus, the graph $H_\text{ind}$ is a loop-threshold graph. We survey known results for maximizing the number of homomorphisms into small loop-threshold image graphs. The only extremal homomorphism problem with a loop-threshold image graph on at most three vertices not yet solved is $H_\text{ind}\cup E_1$, where extremal graphs are the union of a lex graph and an empty graph. The only question that remains is the size of the lex component of the extremal graph. While we cannot give an exact answer for every number of vertices and edges, we establish the significance of and give bounds on $\ell(m)$, the number of vertices in the lex component of the extremal graph with $m$ edges and at least $m+1$ vertices.

Complexity of planar signed graph homomorphisms to cycles

We study homomorphism problems of signed graphs from a computational point of view. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept when studying signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. The concept of s-homomorphisms has been extensively studied, and a full complexity classification (polynomial or NP-complete) for s-homomorphism to a fixed target signed graph has recently been obtained. Nevertheless, such a dichotomy is not known when we restrict the input graph to be planar, not even for non-signed graph homomorphisms.We show that deciding whether a (non-signed) planar graph admits a homomorphism to the square Ct2 of a cycle with t⩾6, or to the circular clique K4t∕(2t−1) with t⩾2, are NP-complete problems. We use these results to show that deciding whether a planar signed graph admits an s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar s-homomorphism problem with any signed cycle as a target.We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar s-homomorphism problems to signed cycles remain NP-complete even for inputs of maximum degree 3 (except for the case of unbalanced 4-cycles, for which we show this for maximum degree 4). We also show that for a given integer g, the problem for signed bipartite planar inputs of girth g is either trivial or NP-complete.

Subexponential algorithms for variants of the homomorphism problem in string graphs

We consider subexponential algorithms finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy: if H has no two vertices sharing two common neighbors, then the problem can be solved in time 2O(n2/3log⁡n), otherwise there is no subexponential algorithm, assuming the ETH. Then we consider locally constrained homomorphisms. We show that for each target graph H, the locally injective and locally bijective homomorphism problems can be solved in time 2O(nlog⁡n) in string graphs. For locally surjective homomorphisms we show a dichotomy for H being a path or a cycle. If H is P3 or C4, then the problem can be solved in time 2O(n2/3log3/2⁡n) in string graphs; otherwise, assuming the ETH, there is no subexponential algorithm. As corollaries, we obtain new results concerning the complexity of homomorphism problems in Pt-free graphs.

Complexity of correspondence [formula omitted]-colourings

Correspondence homomorphisms generalize standard homomorphisms as well as correspondence colourings (also known as DP-colourings). For a fixed target graph H, we study the problem of deciding whether an input graph G, with each edge labelled by a pair of permutations of V(H), admits a homomorphism to H ‘corresponding’ to the labels. Homomorphisms to H are called H-colourings, and we employ the similar term correspondence H-colourings for correspondence homomorphisms to H.We classify the complexity of this problem as a function of the fixed graph H. It turns out that there is dichotomy — each of the problems is polynomial-time solvable or NP-complete. While most graphs H yield NP-complete problems, there are interesting cases of graphs H for which the problem can be solved in polynomial time by Gaussian elimination.We also classify the complexity of the analogous correspondence list homomorphism problems, and also the complexity of a bipartite version of both problems. We give detailed proofs for the case when H is reflexive, and, for the record, sketch the remaining proofs.