Minimum Cost Homomorphisms to Reflexive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$such that$uv\in A(G)$implies$f(u)f(v)\in A(H)$. If moreover each vertex$u \in V(G)$is associated with costs$c_i(u), i \in V(H)$, then the cost of a homomorphism$f$is$\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph$H, the {\em minimum cost homomorphism problem} for H$, denoted MinHOM($H$), is the following problem. Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, and an integer$k$, decide if$G$admits a homomorphism to$H$of cost not exceeding$k. We focus on the minimum cost homomorphism problem for {\em reflexive} digraphs H$(every vertex of$H$has a loop). It is known that the problem MinHOM($H$) is polynomial time solvable if the digraph$H has a {\em Min-Max ordering}, i.e., if its vertices can be linearly ordered by <$so that$i<j, s<r$and$ir, js \in A(H)$imply that$is \in A(H)$and$jr \in A(H)$. We give a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering; our characterization implies a polynomial time test for the existence of a Min-Max ordering. Using this characterization, we show that for a reflexive digraph$H$ which does not admit a Min-Max ordering, the minimum cost homomorphism problem is NP-complete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs

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.Comment: 12 pages, STACS 201

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$such that$uv\in A(G)$implies$f(u)f(v)\in A(H)$. If, moreover, each vertex$u \in V(G)$is associated with costs$c_i(u), i \in V(H)$, then the cost of a homomorphism$f$is$\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph$H$, the minimum cost homomorphism problem for$H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, decide 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 such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes

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 which remain of open complexity

Conservative constraint satisfaction re-revisited

Conservative constraint satisfaction problems (CSPs) constitute an important particular case of the general CSP, in which the allowed values of each variable can be restricted in an arbitrary way. Problems of this type are well studied for graph homomorphisms. A dichotomy theorem characterizing conservative CSPs solvable in polynomial time and proving that the remaining ones are NP-complete was proved by Bulatov in 2003. Its proof, however, is quite long and technical. A shorter proof of this result based on the absorbing subuniverses technique was suggested by Barto in 2011. In this paper we give a short elementary prove of the dichotomy theorem for the conservative CSP

Testing List H-Homomorphisms

Let $H$ be an undirected graph. In the List $H$-Homomorphism Problem, given an undirected graph $G$ with a list constraint $L(v) \subseteq V(H)$ for each variable $v \in V(G)$, the objective is to find a list $H$-homomorphism $f:V(G) \to V(H)$, that is, $f(v) \in L(v)$ for every $v \in V(G)$ and $(f(u),f(v)) \in E(H)$ whenever $(u,v) \in E(G)$. We consider the following problem: given a map $f:V(G) \to V(H)$ as an oracle access, the objective is to decide with high probability whether $f$ is a list $H$-homomorphism or \textit{far} from any list $H$-homomorphisms. The efficiency of an algorithm is measured by the number of accesses to $f$. In this paper, we classify graphs $H$ with respect to the query complexity for testing list $H$-homomorphisms and show the following trichotomy holds: (i) List $H$-homomorphisms are testable with a constant number of queries if and only if $H$ is a reflexive complete graph or an irreflexive complete bipartite graph. (ii) List $H$-homomorphisms are testable with a sublinear number of queries if and only if $H$ is a bi-arc graph. (iii) Testing list $H$-homomorphisms requires a linear number of queries if $H$ is not a bi-arc graph

Building blocks for the variety of absolute retracts

AbstractGiven a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k⩾2 we describe a collection of graphs that generate the variety ARk of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that ARk⊂NUFk+1, where NUFk+1 is the variety of graphs that admit a near unanimity function of arity k+1