List Homomorphism Research Articles

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.

Min Orderings and List Homomorphism Dichotomies for Graphs and Signed Graphs

Min Orderings and List Homomorphism Dichotomies for Graphs and Signed Graphs

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).

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.

Correspondence Homomorphisms to Reflexive Graphs

Correspondence homomorphisms are a common generalization of homomorphisms and of correspondence colourings. For a fixed reflexive target graph H, the problem is to decide whether an input graph G, with each edge labeled by a pair of permutations of V(H), admits a homomorphism to H ‘corresponding’ to the labels. We classify the complexity of this problem as a function of H. It turns out that there is dichotomy—each of the problems is polynomial or NP-complete. While most graphs H yield NP-complete problems, there is an interesting polynomial case when the problem can be solved by Gaussian elimination. We also classify the complexity of the analogous correspondence list homomorphism problems.

List H-Coloring a Graph by Removing Few Vertices

In the deletion version of the list homomorphism problem, we are given graphs G and H, a list $$L(v)\subseteq V(H)$$ for each vertex $$v\in V(G)$$ , and an integer k. The task is to decide whether there exists a set $$W \subseteq V(G)$$ of size at most k such that there is a homomorphism from $$G {\setminus } W$$ to H respecting the lists. We show that DL-Hom( $${H}$$ ), parameterized by k and |H|, is fixed-parameter tractable for any $$(P_6,C_6)$$ -free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom( $${H}$$ ) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. (Combinatorica 19(4):487–505, 1999), a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

On Maltsev Digraphs

We study digraphs preserved by a Maltsev operation: Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing in this way that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. We then generalize results from Kazda (2011) to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a $O(|V_G|^4)$-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation, and relate them with series parallel digraphs.

On List Coloring and List Homomorphism of Permutation and Interval Graphs

List coloring is an NP-complete decision problem even if the total number of colors is three. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving list coloring of permutation graphs with a bounded total number of colors. More generally, we give a polynomial-time algorithm that solves the list-homomorphism problem to any fixed target graph for a large class of input graphs, including all permutation and interval graphs.

The complexity of surjective homomorphism problems—a survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of whose complexity remains open.

Generalising and dualising the third list-homomorphism theorem

The third list-homomorphism theorem says that a function is a list homomorphism if it can be described as an instance of both a foldr and a foldl . We prove a dual theorem for unfolds and generalise both theorems to trees: if a function generating a list can be described both as an unfoldr and an unfoldl , the list can be generated from the middle, and a function that processes or builds a tree both upwards and downwards may independently process/build a subtree and its one-hole context. The point-free, relational formalism helps to reveal the beautiful symmetry hidden in the theorem.

Interval graphs, adjusted interval digraphs, and reflexive list homomorphisms

Interval graphs admit linear-time recognition algorithms and have several elegant forbidden structure characterizations. Interval digraphs can also be recognized in polynomial time, and they admit a characterization in terms of incidence matrices. Nevertheless, they do not have a known forbidden structure characterization or low-degree polynomial-time recognition algorithm.We introduce a new class of ‘adjusted interval digraphs’. By contrast, for these digraphs we exhibit a natural forbidden structure characterization, in terms of a novel structure which we call an ‘invertible pair’. Our characterization also yields a low-degree polynomial-time recognition algorithm of adjusted interval digraphs.It turns out that invertible pairs are also useful for undirected interval graphs, and our result yields a new forbidden structure characterization of interval graphs. In fact, it can be shown to be a natural link proving the equivalence of some known characterizations of interval graphs—the theorems of Lekkerkerker and Boland, and of Fulkerson and Gross.In addition, adjusted interval digraphs naturally arise in the context of list homomorphism problems. If H is a reflexive undirected graph, the list homomorphism problem LHOM(H) is polynomial if H is an interval graph, and NP-complete otherwise. If H is a reflexive digraph, LHOM(H) is polynomial if H is an adjusted interval graph, and we conjecture that it is also NP-complete otherwise. We show that our results imply the conjecture in two important cases.

The Complexity of the List Homomorphism Problem for Graphs

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H, the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

Dichotomy for tree-structured trigraph list homomorphism problems

Trigraph list homomorphism problems, also known as list matrix partition problems, generalize graph list colouring and digraph list homomorphism problems. While digraph list homomorphism problems enjoy a dichotomy (each problem is NP -complete or polynomial time solvable), such dichotomy is not necessarily expected for trigraph list homomorphism problems, and in the few cases where dichotomy has been proved, for small trigraphs, the progress has been slow. In this paper, we prove dichotomy for trigraph list homomorphism problems where the underlying graph of the trigraph is a tree. In fact, we show that for these trigraphs the trigraph list homomorphism problem is polynomially equivalent to a related digraph list homomorphism problem. The result can be extended to a larger class of trigraphs, and we illustrate the extension on trigraph cycles.

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.

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.

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.

Bi‐arc graphs and the complexity of list homomorphisms

AbstractGiven graphs G, H, and lists L(v) ⊆ V(H), v ε V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) → V(H) such that uv ε E(G) implies f(u)f(v) ε E(H), and f(v) ε L(v) for all v ε V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) ⊆ V(H), v ε V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP‐complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP‐complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi‐arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi‐arc graph, and is NP‐complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61–80, 2003

Abstraction and performance in the design of parallel programs: an overview of the SAT approach

SAT stands for Stages And Tranformations and is the name of an approach to the high-level, performance-directed design of parallel programs. The target programs obtained with this approach are sequences of internally parallel stages, i.e., they fall within the SPMD model. Formal program transformations are used for deriving each parallel stage and optimizing the combination of several stages. The main advantage of the SAT approach is the comparatively high level of abstraction at which performance-relevant design decisions can be made. One consequence is that choices become much clearer than at the code level, enhancing comparability; another is that there is an increased potential for automation of the programming process. This paper summarizes the approach and assesses its usefulness, with emphasis on one basic parallel programming pattern, the list homomorphism, which captures the divide-and-conquer paradigm. We survey the transformational theory, provide a range of practical examples, and discuss the potential for automation and also the demands made on application programmers and implementers.