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

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

Hereditarily hard H-colouring problems

AbstractLet H be a graph (respectively digraph) whose vertices are called ‘colours’. An H-colouring of a graph (respectively digraph) G is an assignment of these colours to the vertices of G so that if u is adjacent to v in G, then the colour of u is adjacent to the colour of v in H. We continue the study of the complexity of the H-colouring problem ‘Does a given graph (respectively digraph) admit an H-colouring?’. For graphs it was proved that the H-colouring problem is NP-complete whenever H contains an odd cycle, and is polynomial for bipartite graphs. For directed graphs the situation is quite different, as the addition of an edge to H can result in the complexity of the H-colouring problem shifting from NP-complete to polynomial. In fact, there is not even a plausible conjecture as to what makes directed H-colouring problems difficult in general. Some order may perhaps be found for those digraphs H in which each vertex has positive in-degree and positive out-degree. In any event, there is at least, in this case, a conjecture of a classification by complexity of these directed H-colouring problems. Another way, which we propose here, to bring some order to the situation is to restrict our attention to those digraphs H which, like odd cycles in the case of graphs, are hereditarily hard, i.e., are such that the H′-colouring problem is NP-hard for any digraph H′ containing H as a subdigraph. After establishing some properties of the digraphs in this class, we make a conjecture as to precisely which digraphs are hereditarily hard. Surprisingly, this conjecture turns out to be equivalent to the one mentioned earlier. We describe several infinite families of hereditarily hard digraphs, and identify a family of digraphs which are minimal in the sense that it would be sufficient to verify the conjecture for members of that family

Algebra and the Complexity of Digraph CSPs: a Survey

We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems

Multiplicativity of acyclic digraphs

AbstractA homomorphism of a digraph to another digraph is an edge preserving vertex mapping. A digraph W is said to be multiplicative if the set of digraphs which cannot be homomorphically mapped to W is closed under categorical product. We discuss the necessary conditions for a digraph to be multiplicative. Our main result is that almost all acyclic digraphs which have a Hamiltonian path are nonmultiplicative. We conjecture that almost all digraphs are nonmultiplicative