On the Complexity of the Minimum Cost Homomorphism Problem for Reflexive Multipartite Tournaments

For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ 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 HOMP($H$). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism problem is defined as follows. If each vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$ and denote it as MinHOMP($H$). The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D), i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. In a recent paper, we posed a problem of characterizing polynomial time solvable and NP-hard cases of the minimum cost homomorphism problem for acyclic multipartite tournaments with possible loops (w.p.l.). In this paper, we solve the problem for reflexive multipartite tournaments and demonstrate a considerate difficulty of the problem for the whole class of multipartite tournaments w.p.l. using, as an example, acyclic 3-partite tournaments of order 4 w.p.l.\footnote{This paper was submitted to Discrete Mathematics on April 6, 2007}