Homomorphism Problem Research Articles

Homomorphisms of random paths

We study homomorphisms between randomly directed paths and give estimates on the probability of the existence of such a homomorphism. We show that a random path is a core with a positive probability. We apply our results in the investigation of homomorphism dualities, the most natural situation when a homomorphism (or Constraint Satisfaction) problem is in coNP.

The complexity of the minimum cost homomorphism problem for semicomplete digraphs with possible loops

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM( H ). An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. If each vertex u ∈ V ( D ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( D ) c f ( u ) ( u ) . For each fixed digraph H , we have the minimum cost homomorphism problem for H and denote it as MinHOM( H ). The problem is to decide, for an input graph D with costs c i ( u ) , u ∈ V ( D ) , i ∈ V ( H ) , whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM( H ) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM( H ) were proved when H is a semicomplete digraph Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12,11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in Gutin and Kim (2008) [9].

Adjusted Interval Digraphs

Abstract Interval digraphs were introduced by West et al. They can be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, they do not have a forbidden structure characterization nor a low-degree polynomial recognition algorithm. We introduce a new class of ‘adjusted interval digraphs,’ obtained by a slight change in the definition. We show that, by contrast, these digraphs have a natural forbidden structure characterization, parallel to a characterization for undirected graphs, and admit a simple recognition algorithm. We relate adjusted interval digraphs to a list homomorphism problem. Each digraph H defines a corresponding list homomorphism problem L-HOM(H). We observe that if H is an adjusted interval digraph, then the problem L-HOM(H) is polynomial time solvable, and conjecture that for all other reflexive digraphs H the problem L-HOM(H) is NP-complete. We present some preliminary evidence for the conjecture, including a proof for the special case of semi-complete digraphs.

Colouring, constraint satisfaction, and complexity

Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity — the k -colouring problem is polynomial-time solvable when k ≤ 2 , and NP-complete when k ≥ 3 . Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction. However, for general constraint satisfaction problems, dichotomy has only been conjectured. Although the conjecture remains unproven to this day, it has been driving much of the theoretical research on constraint satisfaction problems, which combines methods of logic, universal algebra, analysis, and combinatorics. Currently, this is a very active area of research, and it is our goal here to present some of the recent developments, updating some of the information in existing books and surveys, while focusing on both the mathematical and the computational aspects of the theory. Given the level of activity, we are only able to survey a fraction of the new work, with emphasis on our own areas of interest.

Extension problems with degree bounds

We have proved in an earlier paper that the complexity of the list homomorphism problem, to a fixed graph H , does not change when the input graphs are restricted to have bounded degrees (except in the trivial case when the bound is two). By way of contrast, we show in this paper that the extension problem, again to a fixed graph H , can, in some cases, become easier for graphs with bounded degrees.

Minimum cost homomorphisms to semicomplete multipartite digraphs

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . For a fixed directed or undirected graph H and an input graph D , the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a real-world problem in defence logistics and was introduced recently by the authors and M. Tso. Suppose we are given a pair of digraphs D , H and a cost c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) . The cost of a homomorphism f of D to H is ∑ u ∈ V ( D ) c f ( u ) ( u ) . Let H be a fixed digraph. The minimum cost homomorphism problem for H , MinHOMP( H ), is stated as follows: For input digraph D and costs c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) , verify whether there is a homomorphism of D to H and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of MinHOMP ( H ) when H is a semicomplete digraph. In this paper we extend the classification to semicomplete k -partite digraphs, k ≥ 3 , and obtain such a classification for bipartite tournaments.

A dichotomy for minimum cost graph homomorphisms

For graphs G and H , a mapping f : V ( G ) → V ( H ) is a homomorphism of G to H if u v ∈ E ( G ) implies f ( u ) f ( v ) ∈ E ( H ) . If, moreover, each vertex u ∈ V ( G ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( G ) c f ( u ) ( u ) . For each fixed graph H , we have the minimum cost homomorphism problem , written as MinHOM ( H ) . The problem is to decide, for an input graph G with costs c i ( u ) , u ∈ V ( G ) , i ∈ V ( H ) , whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H , with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM ( H ) is polynomial time solvable. In all other cases the problem MinHOM ( H ) is NP-hard. This solves an open problem from an earlier paper.

Minimum Cost Homomorphisms to Semicomplete Bipartite Digraphs

For digraphs D and H, a mapping $f:V(D)\rightarrow V(H)$ is a homomorphism of D to H if $uv\in A(D)$ implies $f(u)f(v)\in A(H)$. If, moreover, each vertex $u\in V(D)$ is associated with costs $c_i(u)$, $i\in V(H)$, then the cost of the homomorphism f is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs $c_i(u)$, $u\in V(D)$, $i\in V(H)$, whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a k-Min-Max ordering of digraphs.

On the restricted homomorphism problem

The restricted homomorphism problem RHP ( H , Y ) asks: given an input digraph G and a homomorphism g : G → Y , does there exist a homomorphism f : G → H ? We prove that if H is hereditarily hard and Y ↛ H , then RHP ( H , Y ) is NP-complete. Since non-bipartite graphs are hereditarily hard, this settles in the affirmative a conjecture of Hell and Nešetřil.

On counting homomorphisms to directed acyclic graphs

It is known that if P and NP are different then there is an infinite hierarchy of different complexity classes that lie strictly between them. Thus, if P ≠ NP, it is not possible to classify NP using any finite collection of complexity classes. This situation has led to attempts to identify smaller classes of problems within NP where dichotomy results may hold: every problem is either in P or is NP-complete. A similar situation exists for counting problems. If P ≠#P, there is an infinite hierarchy in between and it is important to identify subclasses of #P where dichotomy results hold. Graph homomorphism problems are a fertile setting in which to explore dichotomy theorems. Indeed, Feder and Vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their long-standing dichotomy conjecture for all constraint satisfaction problems. In this article, we give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs. Let H be a fixed directed acyclic graph. The problem is, given an input digraph G , determine how many homomorphisms there are from G to H . We give a graph-theoretic classification, showing that for some digraphs H , the problem is in P and for the rest of the digraphs H the problem is #P-complete. An interesting feature of the dichotomy, which is absent from previously known dichotomy results, is that there is a rich supply of tractable graphs H with complex structure.

Density and power graphs in graph homomorphism problem

We introduce two necessary conditions for the existence of graph homomorphisms based on the concepts of density and power graph. As corollaries, we obtain a lower bound for the fractional chromatic number, and we set forward elementary proofs of the facts that the circular chromatic number of the Petersen graph is equal to three and the fact that the Coxeter graph is a core.

Grad and classes with bounded expansion II. Algorithmic aspects

Classes of graphs with bounded expansion have been introduced in [J. Nešetřil, P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, O. Delmas (Eds.), 7th International Colloquium on Graph Theory, in: Electronic Notes in Discrete Mathematics, vol. 22, Elsevier (2005), pp. 101–106; J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)]. They generalize classes with forbidden topological minors (i.e. classes of graphs having no subgraph isomorphic to the subdivision of some graph in a forbidden family), and hence both proper minor closed classes and classes with bounded degree. For any class with bounded expansion C and any integer p there exists a constant N(C,p) so that the vertex set of any graph G∈C may be partitioned into at most N(C,p) parts, any i≤p parts of them induce a subgraph of tree-width at most (i−1) [J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)] (actually, of tree-depth [J. Nešetřil, P. Ossona de Mendez, Tree depth, subgraph coloring and homomorphism bounds, European Journal of Combinatorics 27 (6) (2006) 1022–1041] at most i, which is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion III. Restricted dualities, European Journal of Combinatorics (2005) (submitted for publication)]. We give here a simple algorithm for computing such partitions and prove that if we restrict the input graph to some fixed class C with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed C and p). This result is applied to get a linear time algorithm for the subgraph isomorphism problem with fixed pattern and input graphs in a fixed class with bounded expansion. More generally, let ϕ be a first-order logic sentence. We prove that any fixed graph property of type may be decided in linear time for input graphs in a fixed class with bounded expansion. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. Also, extending several earlier results, we show that a class of graphs has sublinear separators if it has sub-exponential expansion. This result is best possible in general.

The complexity of homomorphism and constraint satisfaction problems seen from the other side

We give a complexity theoretic classification of homomorphism problems for graphs and, more generally, relational structures obtained by restricting the left hand side structure in a homomorphism. For every class C of structures, let HOM(C,−) be the problem of deciding whether a given structure A ∈C has a homomorphism to a given (arbitrary) structure ß. We prove that, under some complexity theoretic assumption from parameterized complexity theory, HOM(C,−) is in polynomial time if and only if C has bounded tree width modulo homomorphic equivalence. Translated into the language of constraint satisfaction problems, our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems. Translated into the language of database theory, it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases.

Bounded width problems and algebras

Let \({\mathcal A}\) be finite relational structure of finite type, and let CSP \({(\mathcal A)}\) denote the following decision problem: if \({\mathcal I}\) is a given structure of the same type as \({\mathcal A}\) , is there a homomorphism from \({\mathcal I}\) to \({\mathcal A}\)? To each relational structure \({\mathcal A}\) is associated naturally an algebra \({\mathbb A}\) whose structure determines the complexity of the associated decision problem. We investigate those finite algebras arising from CSP’s of so-called bounded width, i.e., for which local consistency algorithms effectively decide the problem. We show that if a CSP has bounded width then the variety generated by the associated algebra omits the Hobby-McKenzie types 1 and 2. This provides a method to prove that certain CSP’s do not have bounded width. We give several applications, answering a question of Nesetřil and Zhu [26], by showing that various graph homomorphism problems do not have bounded width. Feder and Vardi [17] have shown that every CSP is polynomial-time equivalent to the retraction problem for a poset we call the Feder − Vardi poset of the structure. We show that, in the case where the structure has a single relation, if the retraction problem for the Feder-Vardi poset has bounded width then the CSP for the structure also has bounded width. This is used to exhibit a finite order-primal algebra whose variety admits type 2 but omits type 1 (provided P ≠ NP).

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.

List homomorphisms of graphs with bounded degrees

In a series of papers, we have classified the complexity of list homomorphism problems. Here, we investigate the effect of restricting the degrees of the input graphs. It turns out that the complexity does not change (except when the degree bound is two). We obtain similar results on restricting the size of the lists. We contrast these results with facts about variants of the list homomorphism problem, where restricting the degrees can have an important effect on the complexity.

The structure of bi-arc trees

Bi-arc graphs generalize (reflexive) interval graphs and those (irreflexive) bipartite graphs whose complements are circular arc graphs. They are relevant for the so-called list homomorphism problem: when H is a bi-arc graph, the problem is polynomial time solvable, otherwise it is NP-complete. Bi-arc graphs have a forbidden structure characterization, and can be recognized in polynomial time. More importantly for this paper, bi-arc graphs can be characterized by the existence of a conservative majority function. (This function plays an important role in proving the correctness of a polynomial time list homomorphism algorithm.) The forbidden structure theorem for bi-arc graphs is quite complex, and the existence of a conservative majority function is proved without giving an explicit description of it. In this note we focus on bi-arc graphs that are trees (with loops allowed). We describe the structure of bi-arc trees, and give a simple forbidden subtree characterization. Based on this structure theorem, we are able to explicitly describe the conservative majority functions.

Minimum cost and list homomorphisms to semicomplete digraphs

For digraphs D and H, a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if uv ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected input graph D admits a homomorphism to H . The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V ( D ) is assigned a set L x of possible colors (vertices of H). The following optimization version of these decision problems generalizes the list homomorphism problem and was introduced in Gutin et al. [Level of repair analysis and minimum cost homomorphisms of graphs, Discrete Appl. Math., to appear], where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs D , H and a positive integral cost c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) . The cost of a homomorphism f of D to H is ∑ u ∈ V ( D ) c f ( u ) ( u ) . For a fixed digraph H, the minimum cost homomorphism problem for H is stated as follows: for an input digraph D and costs c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) , verify whether there is a homomorphism of D to H and, if one exists, find such a homomorphism of minimum cost. We obtain dichotomy classifications of the computational complexity of the list homomorphism and minimum cost homomorphism problems, when H is a semicomplete digraph (digraph in which there is at least one arc between any two vertices). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when H is a semicomplete digraph: both problems are polynomial solvable if H has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for the minimum cost homomorphism problem is different: the problem is polynomial time solvable if H is acyclic or H is a cycle of length 2 or 3; otherwise, the problem is NP-hard.

Level of repair analysis and minimum cost homomorphisms of graphs

Level of repair analysis (LORA) is a prescribed procedure for defense logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORA-BR, Barros [The optimisation of repair decisions using life-cycle cost parameters. IMA J. Management Math. 9 (1998) 403–413] and Barros and Riley [A combinatorial approach to level of repair analysis, European J. Oper. Res. 129 (2001) 242–251] developed certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NP-hard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that our result for LORA-BR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable.

Complexities of Homomorphism and Isomorphism for Definite Logic Programs

A homomorphism ϕ of logic programs from P to P' is a function mapping Atoms(P) to Atoms(P') and it preserves complements and program clauses. For each definite program clause a ← a1,...,an ∈ P it implies that ϕ(a) ← ϕ(a1),..., ϕ(an) is a program clause of P'. A homomorphism ϕ is an isomorphism if ϕ is a bijection. In this paper, the complexity of the decision problems on homomorphism and isomorphism for definite logic programs is studied. It is shown that the homomorphism problem (HOM-LP) for definite logic programs is NP-complete, and the isomorphism problem (ISO-LP) is equivalent to the graph isomorphism problem (GI).