## 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 ... |

236 | Digraphs: Theory, Algorithms and Applications,” 2nd Edition
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Citation Context ...r x dominates y or y dominates x or both. A tournament is a semicomplete digraph with no 2-cycle. Semicomplete digraphs and, in particular, tournaments are well-studied in graph theory and algorithms =-=[4]-=-. A digraph G ′ is the dual of a digraph G if G ′ is obtained from G by changing orientations of all arcs. For digraphs D and H, a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) impl... |

216 |
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Citation Context ...ter Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, anders@cs.rhul.ac.uk 1s1 Introduction For excellent introductions to homomorphisms in directed and undirected graphs, see =-=[19, 21]-=-. In this paper, directed (undirected) graphs have no parallel arcs (edges) or loops. The vertex (arc) set of a digraph G is denoted by V (G) (A(G)). The vertex (edge) set of an undirected graph G is ... |

88 |
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- 2003
(Show Context)
Citation Context ...g-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 digraph H, LHOMP(H) is either polynomial time solvable or NP-complete. The same result for HOMP(H) is conjectured, see, e.g., [19, 21]. If this conjecture holds, it wil... |

66 |
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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 =-=[2, 3, 6, 14, 23]-=-), then the second approach is useful. The proof of Theorem 2.5 using the first approach is significantly shorter than that using the second approach. However, we are aware of some cases of digraphs H... |

57 |
On the complexity of H-colouring
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(Show Context)
Citation Context ... ∈ Lv for each v ∈ V (D). The problems HOMP(H) and LHOMP(H) have been studied for several families of directed and undirected graphs H, see, e.g., [19, 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 pol... |

30 | List homomorphisms and circular arc graphs
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(Show Context)
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... |

29 | A maximal tractable class of soft constraints
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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... |

28 |
Commitment and Community
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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
- Feder, Hell, et al.
- 1999
(Show Context)
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... |

26 | Tso M. Level of repair analysis and minimum cost homomorphisms of graphs
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(Show Context)
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... |

22 | Algorithmic aspects of graph homomorphisms
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- 2003
(Show Context)
Citation Context ...ter Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, anders@cs.rhul.ac.uk 1s1 Introduction For excellent introductions to homomorphisms in directed and undirected graphs, see =-=[19, 21]-=-. In this paper, directed (undirected) graphs have no parallel arcs (edges) or loops. The vertex (arc) set of a digraph G is denoted by V (G) (A(G)). The vertex (edge) set of an undirected graph G is ... |

17 |
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(Show Context)
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 |
A coloring problem for weighted graphs
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- 1997
(Show Context)
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 |
Graph colourings
- Gutjahr
- 1991
(Show Context)
Citation Context ...her vertices in V (G) are assigned color k + 1). ⋄ Interestingly, the problem HOMP(H ′ ) for H ′ (especially, with k = 3) defined in Lemma 4.2 is well known to be polynomial time solvable (see, e.g., =-=[5, 13, 17]-=-). The following lemma allows us to prove that MaxHOMP(H) and MinHOMP(H) are NP-hard when MaxHOMP(H ′ ) and MinHOMP(H ′ ) are NP-hard for an induced subdigraph H ′ of H. Lemma 4.3 Let H ′ be an induce... |

13 |
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- 1972
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Citation Context ...ving VCSP(Φ) we will determine whether HOM(H) �= ∅, and find an optimal h ∈ HOM(H), if HOM(H) �= ∅. ⋄ The second approach is based on Theorem 2.3, whose part (i) is a well-known assertion, see, e.g., =-=[18]-=- and Ex. 7 in Ch. 2 of [21]. It appears that Theorem 2.5 is the first nontrivial application of Theorem 2.3. The homomorphic product of digraphs D and H is an undirected graph D ⊗ H defined as follows... |

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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 =-=[2, 3, 6, 14, 23]-=-), then the second approach is useful. The proof of Theorem 2.5 using the first approach is significantly shorter than that using the second approach. However, we are aware of some cases of digraphs H... |

12 |
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Citation Context ...⋄ A labeling 1, 2, . . . , p of the vertices of H satisfies the X-underbar property if for any pair (i, k), (j, s) of arcs in H, we have (min{i, j}, min{k, s}) ∈ A(H). This property was introduced in =-=[18]-=- where it was used to prove that HOMP(H) is polynomial time solvable when H is an oriented path. So, it would be natural to call the property (SM) the X-bar & X-underbar property. The second approach ... |

9 |
Augmenting graphs for independent sets
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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 =-=[2, 3, 6, 14, 23]-=-), then the second approach is useful. The proof of Theorem 2.5 using the first approach is significantly shorter than that using the second approach. However, we are aware of some cases of digraphs H... |

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Citation Context |

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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... |

7 |
Soft constraints: Complexity and multimorphisms
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- 2003
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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 ... 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]-=-, where weighted Max CSP is investigated. The problem VCSP and some of its special cases generalize MinHOMP(H). We consider VCSP in the next section and demonstrate that an important result on VCSP de... |

2 | Maximum weighted independent sets on transitive graphs and applications - Kagaris, Tragoudas - 1999 |

1 |
On the number of maximal independent sets in graphs from hereditary classes. In: Combinatorial-algebraic methods in discrete optimization, Univ. of Nizhny Novgorod
- Alekseev
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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 /∈ ... |