Minimum cost and list homomorphisms to semicomplete digraphs

Abstract

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 graph input digraph 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 was introduced in [12], 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 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, MinHOMP(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 it exists, find such a homomorphism of minimum cost.We obtain dichotomy classifications of the computational complexity of the list homomorphism problem and MinHOMP(H), when H is a semicomplete digraph (a digraph in which every two vertices have at least one arc between them). 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 MinHOMP(H) 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.

New Plain-Exponential Time Classes for Graph Homomorphism

“…For more on the homomorphism problem in general, we refer to the book by Hell and Nešetřil [16]. Also, in addition to the decision and counting problems considered in this paper, optimization variants have been considered and studied in [14,13,12] and in [18].…”

Minimum Cost Homomorphisms to Reflexive Digraphs

“…If H has a Min-Max ordering, then MinHOM(H) is polynomial time solvable [13] see also [4,26]. Now using our forbidden induced subgraph characterization we can prove that reflexive digraphs H without a Min-Max ordering yield NP-complete MinHOM(H) problems.…”

“…Now using our forbidden induced subgraph characterization we can prove that reflexive digraphs H without a Min-Max ordering yield NP-complete MinHOM(H) problems. Note that we already know that MinHOM(S(H)) is NP-complete if S(H) is not a proper interval graph, and MinHOM(B(H)) is NP-complete if B(H) is not a proper interval bigraph [13]. We begin with a few simple observations.…”

“…It is known that if H admits a Min-Max ordering, then the problem MinHOM(H) is polynomial time solvable [13], see also [4,26]; however, there are digraphs with polynomial MinHOM(H) which do not have Min-Max ordering [14]. For undirected graphs, all H without a Min-Max ordering yield an NP-complete MinHOM(H) [17]; moreoever, having a Min-Max ordering can be characterized by simple forbidden induced subgraphs, and recognized in polynomial time [17].…”

“…Thus, the minimum cost homomorphism problem for graphs has been handled, and interest shifted to directed graphs. The first studies [13,14,15] focused on irreflexive digraphs (no vertex has a loop), where dichotomies has been obtained for digraphs H such that U (H) is a complete or complete multipartite graph. More recently, [11] promoted the study of digraphs with loops allowed; and, in particular, of reflexive digraphs.…”