Extensions of the minimum cost homomorphism problem

  • Authors:

  • Rustem Takhanov

  • Affiliations:

  • Department of Computer and Information Science, Linköping University, Linköping, Sweden

  • Venue:
  • COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
  • Year:
  • 2010

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Abstract

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 (MinHomR) is a set of variables V subject to specified constraints together with a positive weight cvr for each combination of v ∈ V and r ∈ R. The aim is to find a function f : V → D such that f satisfies all constraints and Σv∈V Σr∈R cvrr(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 MinHomR 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 MinHomR is the following statement: if Γ is a constraint language, then MinHomR 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 MinHomR 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