Article

Minimum Cost Homomorphism Dichotomy for Oriented Cycles

Graphs and Combinatorics (Impact Factor: 0.39). 11/2009; 25(4):521-531. DOI: 10.1007/s00373-009-0853-9
Source: DBLP

ABSTRACT

For digraphs D and H, a mapping f : V(D) → V(H) is a homomorphism of D to H if uv ∈ A(D) implies f(u) f(v) ∈ A(H). If, moreover, each vertex u ∈ V(D) is associated with costs c
i
(u), i ∈ V(H), then the cost of the homomorphism f is ∑u ∈V(D)c
f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for
H (abbreviated MinHOM(H)). The problem is to decide, for an input graph D with costs c
i
(u), u ∈ V(D), i ∈ V(H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. We obtain a dichotomy classification for the time complexity of MinHOM(H) when H is an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops.

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    • "M inHom was introduced in [11] where it was motivated by a real-world problem in defence logistics. The question for which directed graphs H the problem M inHom ({H}) is polynomial-time solvable was considered in [8] [9] [10] [11] [12]. "
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    ABSTRACT: In the classical Constraint Satisfaction Problem(CSP) two finite models are given and we are asked to find their homomorphism. In the MinHom problem, besides the models, a set of weighted pairs of elements of this two models is given and the task is to find a homomorphism that maximizes the weight of pairs consistent with the homomorphism, i.e. pairs for which homomorphism maps the first element of the pair to the second element. MinHom can be considered as a generic model for a class of combinatorial optimization problems, one of which is a maximal independent set. It appears naturally in defence logistic and supervised learning. This problem shares a lot of common with the classical CSP. We show that it allows similar algebraic approach to the classification of tractable cases of this problem that connects it with relational and functional clones of multi-valued logic. Using this approach we obtain complete classification of polynomially tractable subcases of MinHom. As a result of this classification we confirm general dichotomy conjecture that was given for various special cases of MinHom in terms of digraph theory[12,11].
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    ABSTRACT: The minimum cost homomorphism problem has arisen as a natural and useful optimization problem in the study of graph (and digraph) coloring and homomorphisms: it unifies a number of other well studied optimization problems. It was shown by Gutin, Rafiey, and Yeo that the minimum cost problem for homomorphisms to a digraph $H$ that admits a so-called extended Min-Max ordering is polynomial time solvable, and these authors conjectured that for all other digraphs $H$ the problem is NP-complete. In a companion paper, we gave a forbidden structure characterization of digraphs that admit extended Min-Max orderings. In this paper, we apply this characterization to prove Gutin's conjecture.
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