  Assume $D$ is a finite set and $R$ is a finite set of functions from $D$ to
the natural numbers. An instance of the minimum $R$-cost homomorphism problem
($MinHom_R$) is a set of variables $V$ subject to specified constraints
together with a positive weight $c_{vr}$ for each combination of $v \in V$ and
$r \in R$. The aim is to find a function $f:V \rightarrow D$ such that $f$
satisfies all constraints and $\sum_{v \in V} \sum_{r \in R} c_{vr}r(f(v))$ is
minimized.
  This problem unifies well-known optimization problems such as the minimum
cost homomorphism problem and the maximum solution problem, and this makes it a
computationally interesting fragment of the valued CSP framework for
optimization problems. We parameterize $MinHom_R\left(\Gamma\right)$ by {\em
constraint languages}, i.e. sets $\Gamma$ of relations that are allowed in
constraints. A constraint language is called {\em conservative} if every unary
relation is a member of it; such constraint languages play an important role in
understanding the structure of constraint problems. The dichotomy conjecture
for $MinHom_R$ is the following statement: if $\Gamma$ is a constraint
language, then $MinHom_R\left(\Gamma\right)$ is either polynomial-time solvable
or NP-complete. For $MinHom$ the dichotomy result has been recently obtained
[Takhanov, STACS, 2010] and the goal of this paper is to expand this result to
the case of $MinHom_R$ with conservative constraint language. For arbitrary $R$
this problem is still open, but assuming certain restrictions on $R$ we prove a
dichotomy. As a consequence of this result we obtain a dichotomy for the
conservative maximum solution problem.
