Minimum Cost Homomorphisms to Reflexive Digraphs

Abstract

For digraphs G and H, a homomorphism of G to H is a mapping f : V (G)→V (H) such that uv ∈ A(G) implies f (u)f (v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs c i (u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f (u) (u). For each fixed digraph H, the 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 ∈ V (G), i ∈ V (H), and an integer k, decide if G admits a homomorphism to H of cost not exceeding k. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as chromatic partition optimization and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for 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 Min-Max ordering, i.e., if its vertices can be linearly ordered by < so that i < j, s < r and ir, js ∈ A(H) imply that is ∈ A(H) and jr ∈ 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, as conjectured by Gutin and Kim. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs.

Colouring, constraint satisfaction, and complexity

“…There has also been much activity for digraphs, where the situation is more complicated [89,91,85,86,87]. In particular, there are polynomial time solvable MinCost CSP's which do not admit a min-max ordering [89], and hence are not covered by a polynomial case of Soft CSP(H).…”

On Some Combinatorial Optimization Problems : Algorithms and Complexity

“…The minimum cost homomorphism problems were introduced by Gutin, Rafiey, Yeo, and Tso [44] and studied in a series of papers that classified the complexity for different classes of binary languages [40][41][42][43]. A complete classification for general constraint languages was obtained by Takhanov in [90].…”

Two new homomorphism dualities and lattice operations

“…We now show that they are polymorphisms of H. Let (a 1 , b 1 ) and (a 2 , b 2 ) be arcs in H. We need to show that (min(a 1 , a 2 ), min(b 1 , b 2 )) and (max(a 1 , a 2 ), max(b 1 , b 2 )) are also arcs in H. We consider the former case, the latter is similar. Assume without loss of generality that min(a 1 , a 2 ) = a Reflexive digraphs that admit polymorphisms min and max with respect to some linear ordering of the vertices were characterised in [19]. Obviously, if H is such a digraph then H u has caterpillar duality.…”