## Minimum Cost Homomorphisms to Semicomplete Multipartite Digraphs

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Citations: | 12 - 8 self |

### BibTeX

@MISC{Gutin_minimumcost,

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

title = {Minimum Cost Homomorphisms to Semicomplete Multipartite Digraphs},

year = {}

}

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### Abstract

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 ci(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. The problem is to decide, for an input graph D with costs ci(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. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a k-Min-Max ordering of digraphs. Key words. homomorphisms, minimum cost homomorphisms, semicomplete bipartite digraphs

### Citations

234 | Digraphs: Theory, Algorithms and Applications
- Bang-Jensen, Gutin
- 2002
(Show Context)
Citation Context ..., v k 2, . . . , v k ℓ(k) Note that if H is a strong digraph in which the greatest common divisor of all cycle lengths is k, then V (H) has a k-partition, k ≥ 2, satisfying (i) (see Theorem 10.5.1 in =-=[1]-=-). A simple example of a digraph having a k-Min-Max ordering is an extension of � Ck. Dichotomy and paper organization. The main result of this paper is the following: Theorem 1.4 Let H be an semicomp... |

215 |
Graphs and homomorphisms
- Hell, Neˇsetˇril
- 2004
(Show Context)
Citation Context ...world problem in defence logistics. We believe it offers a practical and natural model for optimization of weighted homomorphisms. MCH’s special cases include the well-known list homomorphism problem =-=[8, 10]-=- and the general optimum cost chromatic partition problem, which has been intensively studied [7, 11, 12], and has a number of applications, [14, 15]. Minimum cost homomorphisms. For directed or undir... |

67 | The approximability of constraint satisfaction problems
- Khanna, Sudan, et al.
- 2001
(Show Context)
Citation Context ...stead of i ′ . If the new i and i ′ are equal than (b) holds and otherwise we are done by induction. ⋄ The construction used in the following theorem was inspired by somewhat similar constructions in =-=[13]-=- and [2]. Theorem 2.2 If a digraph H has a k-Min-Max ordering, then MinHOM(H) is polynomialtime solvable. Proof: Let H have a k-Min-Max ordering. Let V1, V2, . . . , Vk be defined as in Definition 1.3... |

57 |
On the complexity of H-colouring
- Hell, Nesetril
- 1990
(Show Context)
Citation Context ...or T T − k+1 or � C3, then MinHOM(H) is polynomial time solvable. Otherwise, MinHOM(H) is NP-hard. 6 Further Research In the case of undirected graphs H, the well-known theorem of Hell and Neˇsetˇril =-=[9]-=- on the homomorphism problem implies that MinHOM(H) is NP-hard for each non-bipartite graph H. The authors of [3] obtained a complete dichotomy of the computational complexity of MinHOM(H) when H is u... |

29 | A maximal tractable class of soft constraints
- Cohen, Cooper, et al.
(Show Context)
Citation Context ...i ′ . If the new i and i ′ are equal than (b) holds and otherwise we are done by induction. ⋄ The construction used in the following theorem was inspired by somewhat similar constructions in [13] and =-=[2]-=-. Theorem 2.2 If a digraph H has a k-Min-Max ordering, then MinHOM(H) is polynomialtime solvable. Proof: Let H have a k-Min-Max ordering. Let V1, V2, . . . , Vk be defined as in Definition 1.3. If the... |

26 | Tso M. Level of repair analysis and minimum cost homomorphisms of graphs
- Gutin, Rafiey, et al.
(Show Context)
Citation Context ... that properties of this notion and, in particular, Theorem 2.2 can be used to obtain further results on MCH and its special cases (see below). The minimum cost homomorphism problem was introduced in =-=[6]-=-, where it was motivated by a real-world problem in defence logistics. We believe it offers a practical and natural model for optimization of weighted homomorphisms. MCH’s special cases include the we... |

24 | Complexity results for the optimum cost chromatic partition problem
- Jansen
- 1997
(Show Context)
Citation Context ...on of weighted homomorphisms. MCH’s special cases include the well-known list homomorphism problem [8, 10] and the general optimum cost chromatic partition problem, which has been intensively studied =-=[7, 11, 12]-=-, and has a number of applications, [14, 15]. Minimum cost homomorphisms. For directed or undirected graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v)... |

22 | Algorithmic aspects of graph homomorphisms
- Hell
- 2003
(Show Context)
Citation Context ...world problem in defence logistics. We believe it offers a practical and natural model for optimization of weighted homomorphisms. MCH’s special cases include the well-known list homomorphism problem =-=[8, 10]-=- and the general optimum cost chromatic partition problem, which has been intensively studied [7, 11, 12], and has a number of applications, [14, 15]. Minimum cost homomorphisms. For directed or undir... |

22 |
Finding a maximum planar subset of a set of nets in a channel
- Supowit
- 1987
(Show Context)
Citation Context ...es include the well-known list homomorphism problem [8, 10] and the general optimum cost chromatic partition problem, which has been intensively studied [7, 11, 12], and has a number of applications, =-=[14, 15]-=-. Minimum cost homomorphisms. For directed or undirected graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). Recent treatments of homomorphisms ... |

18 | Minimum cost and list homomorphisms to semicomplete digraphs
- Gutin, Hell, et al.
(Show Context)
Citation Context .... V and U as follows: We orient all edges from V to U and apply the above definition for digraphs. Importance of Min-Max ordering for MinHOM(H) is indicated in the following two theorems. Theorem 1.1 =-=[4]-=- Let a digraph H have a Min-Max ordering. Then MinHOM(H) is polynomial-time solvable. A bipartite graph H with vertices x1, x2, x3, x4, y1, y2, y3 is called a bipartite claw if its edge set E(H) = {x4... |

17 | The optimal cost chromatic partition problem for trees and interval graphs
- Kroon, Sen, et al.
- 1997
(Show Context)
Citation Context ...es include the well-known list homomorphism problem [8, 10] and the general optimum cost chromatic partition problem, which has been intensively studied [7, 11, 12], and has a number of applications, =-=[14, 15]-=-. Minimum cost homomorphisms. For directed or undirected graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). Recent treatments of homomorphisms ... |

16 | Coloring of trees with minimum sum of colors
- Jiang, West
- 1999
(Show Context)
Citation Context ...on of weighted homomorphisms. MCH’s special cases include the well-known list homomorphism problem [8, 10] and the general optimum cost chromatic partition problem, which has been intensively studied =-=[7, 11, 12]-=-, and has a number of applications, [14, 15]. Minimum cost homomorphisms. For directed or undirected graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v)... |

13 | A dichotomy for minimum cost graph homomorphisms
- Gutin, Hell, et al.
(Show Context)
Citation Context ... for the case k ≥ 3 was obtained in [6] (see also Section 5). Our result uses and significantly extends a dichotomy for the computational complexity of MCH for bipartite undirected graphs obtained in =-=[4]-=-. In our previous papers we used properties of an important notion of Min-Max ordering of digraphs. To obtain the dichotomy of this paper, we introduce and study a new notion, a k-Min-Max ordering of ... |

12 |
Minimizing Average Completion of Dedicated Tasks and Interval Graphs
- Halldórsson, Kortsarz, et al.
- 2001
(Show Context)
Citation Context ...on of weighted homomorphisms. MCH’s special cases include the well-known list homomorphism problem [8, 10] and the general optimum cost chromatic partition problem, which has been intensively studied =-=[7, 11, 12]-=-, and has a number of applications, [14, 15]. Minimum cost homomorphisms. For directed or undirected graphs G and H, a mapping f : V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v)... |

10 | Rafiey A. Minimum cost homomorphisms to reflexive digraphs
- Gupta, Hell, et al.
- 2008
(Show Context)
Citation Context ...-Min-Max ordering of digraphs. We believe that properties of this notion and, in particular, Theorem 2.2 can be used to obtain further results on MCH and its special cases. Recent results obtained in =-=[3, 7]-=- and other papers led us to conjecture in [7] that, unless P=NP, MCH is polynomial time solvable only when H admits either a Min-Max ordering or a k-Min-Max ordering for some k ≥ 2. The minimum cost h... |

5 | Minimum Cost Homomorphisms to Proper Interval Graphs and Bigraphs
- Gutin, Hell, et al.
(Show Context)
Citation Context ... for the case k ≥ 3 was obtained in [5] (see also Section 5). Our result uses and significantly extends a dichotomy for the computational complexity of MCH for bipartite undirected graphs obtained in =-=[3]-=-. ∗ Corresponding author. Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, gutin@cs.rhul.ac.uk † School of Computing Science, Simon Fraser University, B... |

4 | Minimum Cost Homomorphism Dichotomy for Oriented Cycles
- Gutin, Rafiey, et al.
(Show Context)
Citation Context ...-Min-Max ordering of digraphs. We believe that properties of this notion and, in particular, Theorem 2.2 can be used to obtain further results on MCH and its special cases. Recent results obtained in =-=[3, 7]-=- and other papers led us to conjecture in [7] that, unless P=NP, MCH is polynomial time solvable only when H admits either a Min-Max ordering or a k-Min-Max ordering for some k ≥ 2. The minimum cost h... |