Extensions of the Minimum Cost Homomorphism Problem

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 (M inHomR) 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 v∈V r∈R cvrr(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 M inHomR (Γ ) 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 M inHomR is the following statement: if Γ is a constraint language, then M inHomR (Γ ) is either polynomial-time solvable or NP-complete. For M inHom 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 M inHomR 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.

The Complexity of Three-Element Min-Sol and Conservative Min-Cost-Hom

“…It was studied in a series of papers before it was completely solved in [18]. The more general version of the problem which we are interested in was introduced in [19]. 1 Methods and Results.…”

“…We obtain a full classification of the complexity of Min-Sol on domains that contain at most three elements. The tractable cases are given by languages that can be solved by a certain linear programming formulation [20] and a new class that is inspired by, and generalises, languages described in [18,19]. A precise classification is given by Theorem 16.…”

“…For conservative Min-Cost-Hom (i.e. Min-Cost-Hom with languages containing all unary crisp cost functions) an almost complete classification (for arbitrary finite domains) was obtained by Takhanov [19]. We are able to remove the extra conditions needed in [19] and provide a full classification for this problem.…”

“…We are able to remove the extra conditions needed in [19] and provide a full classification for this problem. This answers a question raised in [18,19]. The main mathematical tools used througout the paper are from the so-called algebraic approach, see e.g.…”

The Complexity of Valued Constraint Satisfaction Problems

“…In a long series of paper, Gutin et al have classified this problem for various special classes of digraphs [144]. All these partial results have been generalised by Takhanov, who has provided a complete classification in [261], with generalisations in [262]. [176,177], which is a restriction of GRAPH MIN-COST HOMOMORPHISM from Example 1.19, but a generalisation of both GRAPH LIST HOMOMORPHISM from Example 1.20 and MAX-ONES from Example 1.18.…”