TY - GEN

T1 - Extensions of the minimum cost homomorphism problem

AU - Takhanov, Rustem

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-642-14031-0_36

DO - 10.1007/978-3-642-14031-0_36

M3 - Conference contribution

AN - SCOPUS:77955018422

SN - 3642140300

SN - 9783642140303

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 328

EP - 337

BT - Computing and Combinatorics - 16th Annual International Conference, COCOON 2010, Proceedings

T2 - 16th Annual International Computing and Combinatorics Conference, COCOON 2010

Y2 - 19 July 2010 through 21 July 2010

ER -