A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. For a fixed graph $H$, in the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is equipped with a list $L(v) \subseteq V(H)$. We ask if there exists a homomorphism $f$ from $G$ to $H$, in which $f(v) \in L(v)$ for every $v \in V(G)$. Feder, Hell, and Huang [JGT~2003] proved that LHom($H$) is polynomial time-solvable if $H$ is a bi-arc-graph, and NP-complete otherwise.
We are interested in the complexity of the LHom($H$) problem in graphs excluding a copy of some fixed graph $F$ as an induced subgraph. It is known that if $F$ is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom($H$) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs $F$.
If $F$ is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if $H$ is not predacious, then for every fixed $t$ the LHom($H$) problem can be solved in quasi-polynomial time in $P_t$-free graphs. On the other hand, if $H$ is predacious, then there exists $t$, such that LHom($H$) cannot be solved in subexponential time in $P_t$-free graphs.
If $F$ is a subdivided claw, we show a full dichotomy in two important cases: for $H$ being irreflexive (i.e., with no loops), and for $H$ being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive $H$ the LHom($H$) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if $H$ is non-predacious and triangle-free. If $H$ is reflexive, then LHom($H$) cannot be solved in subexponential time whenever $H$ is not a bi-arc graph.
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.
Trigraph list homomorphism problems (also known as list matrix partition problems) have generated recent interest, partly because there are concrete problems that are not known to be polynomial time solvable or NP-complete. Thus while digraph list homomorphism problems enjoy dichotomy (each problem is NP-complete or polynomial time solvable), such dichotomy is not necessarily expected for trigraph list homomorphism problems. However, in this paper, we identify a large class of trigraphs for which list homomorphism problems do exhibit a dichotomy. They consist of trigraphs with a tree-like stru…
The Dichotomy Conjecture for constraint satisfaction problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction between the tractable and intractable cases. Such structural characterization is absent in Bulatov's classification, and Bulatov asked whether one can be found. We provide an answer in the case of digraphs; the technique will apply in a broader context. The key concept we introduce is that of a digr…
For graphs $G,H$, a homomorphism from $G$ to $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the list homomorphism problem, denoted by \textsc{LHom}($H$), we are given a graph $G$ and lists $L: V(G) \to 2^{V(H)}$, and we ask for a homomorphism from $G$ to $H$ which additionally respects the lists $L$. Very recently Okrasa, Piecyk, and Rzążewski [ESA 2020] defined an invariant $i^*(H)$ and proved that under the SETH $\mathcal{O}^*\left (i^*(H)^{\textrm{tw}(G)}\right)$ is the tight complexity bound for \textsc{LHom}($H$), parameterized by the treewidth $\textrm{tw}(G)$ of the instanc…
We show that the set of homomorphisms between two supersolvable groups can be locally list decoded up to the minimum distance of the code, extending the results of Dinur et al who studied the case where the groups are abelian. Moreover, when specialized to the abelian case, our proof is more streamlined and gives a better constant in the exponent of the list size. The constant is improved from about 3.5 million to 105.
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…
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 which remain of open complexity.
We consider a generalization of finding a homomorphism from an input digraph $G$ to a fixed digraph $H$, HOM($H$). In this setting, we are given an input digraph $G$ together with a list function from $G$ to $2^H$. The goal is to find a homomorphism from $G$ to $H$ with respect to the lists if one exists. We show that if the list function is a Maltsev polymorphism then deciding whether $G$ admits a homomorphism to $H$ is polynomial time solvable. In our approach, we only use the existence of the Maltsev polymorphism. Furthermore, we show that deciding whether a relational structure $\mathcal{R…
The codewords of the homomorphism code $\operatorname{aHom}(G,H)$ are the affine homomorphisms between two finite groups, $G$ and $H$, generalizing Hadamard codes. Following the work of Goldreich--Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to $\textsf{mindist}$, the minimum distance of the code. In particular, for the first time, we do not require either $G$ or $H$ to be solvable. Specifically, we demonstrate a $\operatorname{poly}(1/\varepsilon)$ bound on the list siz…
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}…
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by \textsc{LHom}($H$), the instance is a graph $G$, whose every vertex is equipped with a subset of $V(H)$, called list. We ask whether there exists a homomorphism from $G$ to $H$, such that every vertex from $G$ is mapped to a vertex from its list. We study the complexity of the \textsc{LHom}($H$) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the q…
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is assigned with a list $L(v)$ of vertices of $H$. We ask whether there exists a homomorphism $h$ from $G$ to $H$, which respects lists $L$, i.e., for every $v \in V(G)$ it holds that $h(v) \in L(v)$. The complexity dichotomy for LHom($H$) was proven by Feder, Hell, and Huang [JGT 2003]. We are interested in the complexity of the problem, paramete…
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…
We investigate the List $H$-Coloring problem, the generalization of graph coloring that asks whether an input graph $G$ admits a homomorphism to the undirected graph $H$ (possibly with loops), such that each vertex $v \in V(G)$ is mapped to a vertex on its list $L(v) \subseteq V(H)$. An important result by Feder, Hell, and Huang [JGT 2003] states that List $H$-Coloring is polynomial-time solvable if $H$ is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an $n$-vertex instance be…
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first as…
Correspondence homomorphisms are both a generalization of standard homomorphisms and a generalization of correspondence colourings. For a fixed 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, in a sense explained below. 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, t…
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 examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem wh…
Digraphs H for which the list homomorphism problem with template H (LHOM(H)) is in logspace (L) was characterized by Egri et al. (SODA 2014): LHOM(H) is in L if and only if H does not contain a circular N (assuming L is different from NL). Undirected graphs for which LHOM(H) is in L can be characterized in terms forbidden induced subgraphs, and also via a simple inductive construction (Egri et al., STACS 2010). As a consequence, the logspace algorithm in the undirected case is simple and easy to understand. No such forbidden subgraph or inductive characterization, and no such simple and easy-t…
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…
The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is h…
We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, List $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving 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 through a proof that the Surjective $C_6$-Homomorphism problem is NP-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vika…
