Extensions of the minimum cost homomorphism problem

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 ε V and r ε R. The aim is to find a function f:V →D such that f satisfies all constraints and σ _vεV σ ε r ε R c _vr r(f(v)) is maximized. 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 by constraint languages, i.e. sets Γ of relations that are allowed in constraints. A constraint language is called 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 Γ is a constraint language, then MinHom _R 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.