The complexity of the minimum cost homomorphism problem for semicomplete digraphs with possible loops

Abstract

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM( H ). An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. If each vertex u ∈ V ( D ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( D ) c f ( u ) ( u ) . For each fixed digraph H , we have the minimum cost homomorphism problem for H and denote it as MinHOM( H ). The problem is to decide, for an input graph D with costs c i ( u ) , u ∈ V ( D ) , i ∈ V ( H ) , whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM( H ) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM( H ) were proved when H is a semicomplete digraph Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12,11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in Gutin and Kim (2008) [9].