Complexity of the Minimum Cost Homomorphism Problem for Semicomplete Digraphs with Possible Loops

For digraphs $D$ and $H$, a mapping $f:V(D)\text{dom}V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H)\mathrm{.}$ 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 \cite{gutinDAM154a}. 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 $\sum_{u\in V(D)}c_{f(u)}(u)$\mathrm{.}Foreachfixeddigraph$H$, we have the {\em minimum cost homomorphism problem for} $H$ and denote it as MinHOM($H$). The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u\in V(D),i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. 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 \cite{gutinDAM154b}, and a semicomplete multipartite digraph \cite{gutinDAM}. 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 \cite{gutinRMS}.\footnote{This paper was submitted to SIAM J. Discrete Math. on October 27, 2006}