## Minimum Cost and List Homomorphisms to Semicomplete Digraphs

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Venue: | Discrete Appl. Math |

Citations: | 18 - 9 self |

### BibTeX

@ARTICLE{Gutin_minimumcost,

author = {Gregory Gutin and Arash Rafiey and Anders Yeo},

title = {Minimum Cost and List Homomorphisms to Semicomplete Digraphs},

journal = {Discrete Appl. Math},

year = {},

volume = {154},

pages = {890--897}

}

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### 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). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected graph input digraph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (D) is assigned a set Lx of possible colors (vertices of H). The following optimization version of these decision problems was introduced in [16], where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs D, H and a positive cost ci(u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphism f of D to H is � u∈V (D) cf(u)(u). For a fixed digraph H, the minimum cost homomorphism problem for H, MinHOMP(H), is stated as follows: For an input digraph D and costs ci(u) for each u ∈ V (D) and i ∈ V (H), verify whether there is a homomorphism of D to H and, if it exists, find

### Citations

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Citation Context ... result for HOMP(H) is conjectured, see, e.g., [19, 21]. If this conjecture holds, it will imply that the well-known Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi also holds =-=[12]-=-. The authors of [16] introduced an optimization problem on H-colorings for undirected graphs H, MinHOMP(H). The problem is motivated by a problem in defence logistics. Suppose we are given a pair of ... |

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87 |
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Citation Context ..., 21]. A well-known result of Hell and Neˇsetˇril [20] asserts that HOMP(H) for undirected graphs is polynomial time solvable if H is bipartite and it is NP-complete, otherwise. Feder, Hell and Huang =-=[11]-=- proved that LHOMP(H) for undirected graphs is polynomial time solvable if H is a bipartite graph whose complement is a circular arc graph (a graph isomorphic to the intersection graph of arcs on a ci... |

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Citation Context ...or i, we assign every out-neighbor of y color i + 1 modulo k and every in-neighbor of y color i − 1 modulo k. We have HOM(D, � Ck) �= ∅ if and only if no vertex is assigned different colors. M. Green =-=[13]-=- was the first to prove Theorem 3.1 for the case of unicyclic tournaments, but his proof uses polymorphisms (for the definition and results on polymorphisms, see, e.g., [7]). Our proof below is elemen... |

27 |
List homomorphisms and circular arc graphs, Combinatorica 19
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- 1999
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Citation Context ..., 17]. A well-known result of Hell and Neˇsetˇril [16] asserts that HOMP(H) for undirected graphs is polynomial time solvable if H is bipartite and it is NP-complete, otherwise. Feder, Hell and Huang =-=[8]-=- proved that LHOMP(H) for undirected graphs is polynomial time solvable if H is a bipartite graph whose complement is a circular arc graph (a graph isomorphic to the intersection graph of arcs on a ci... |

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Citation Context ...f the homomorphism problem for H, where every vertex x ∈ V (D) is assigned a set Lx of possible colors (vertices of H). The following optimization version of these decision problems was introduced in =-=[16]-=-, where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs D, H and a positive cost ci(u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphi... |

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Citation Context ...) is NP-complete, otherwise. Such a dichotomy classification for all digraphs is unknown and only partial classifications have been obtained; see [21]. For example, Bang-Jensen, Hell and MacGillivray =-=[5]-=- showed that HOMP(H) for semicomplete digraphs H is polynomial time solvable if H has at most one cycle and HOMP(H) is NP-complete, otherwise. Nevertheless, Bulatov [7] managed to prove that for each ... |

13 |
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Citation Context ...p, and the costs are assigned only to edges of H. Thus, MinHOMP(H) and the minimum graph homomorphism problem are rather different problems. Another coloring problem is introduced and investigated in =-=[15]-=-, but it is very different from MinHOMP(H). The maximum cost homomorphism problem MaxHOMP(H) is the same problem as MinHOMP(H), but instead of minimization we consider maximization. Let M be a constan... |

13 |
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Citation Context ...-hardness results are for some special cases of VCSP that do not generalize MinHOMP(H). VCSP extends another optimization problem on H-colorings, the minimum graph homomorphism problem, introduced in =-=[1]-=-. However, the authors of [1] considered only reflexive undirected graphs H, i.e., graphs in which every vertex of H has a loop, and the costs are assigned only to edges of H. Thus, MinHOMP(H) and the... |

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Citation Context ...OMP(H) is NP-hard. This implies that even when H is a unicyclic semicomplete digraph on at least four vertices, MinHOMP(H) is NP-hard (unlike HOMP(H) and LHOMP(H)). Cohen, Cooper, Jeavons and Krokhin =-=[8, 9]-=- considered an optimization version of the well-known constraint satisfaction problem (CSP), the valued CSP (abbreviated VCSP). Special cases of VCSP were studied in several other papers including [10... |

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Citation Context ...H) of undirected graphs. If the last problem is polynomial time solvable (when, for example, F(H) consists of perfect graphs, 2P2-free graphs, claw-free graphs or graphs of other special classes, see =-=[1, 2, 3, 6, 11, 18]-=-), then our approach is useful. The homomorphic product of digraphs D and H is an undirected graph D ⊗ H defined as follows: V (D ⊗H) = {ui : u ∈ V (D), i ∈ V (H)}, E(D ⊗H) = {uivj : uv ∈ A(D), ij /∈ ... |