### Abstract

For digraphs D and H, a homomorphism of D to H is a mapping f V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) 2 A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i 2 V (H), are nonnegative real costs. The cost of the homomorphism f of D to H is Σ u∈V (D) cf(u)(u). The minimum cost homomorphism for a fixed digraph H, denoted by MinHOM(H), asks whether or not an input digraph D, with nonnegative real costs ci(u), u 2 V (D), i 2 V (H), admits a homomorphism f to H and if it admits one, find a homomorphism of minimum cost. The minimum cost homomorphism problem seems to offer a natural and practical way to model many optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. Our interest is in proving a dichotomy for minimum cost homomorphism problem: we would like to prove that for each digraph H, MinHOM(H) is polynomial-time solvable, or NP-hard. We say that H is a digraph with some loops, if H has at least one loop. For reflexive digraphs H (every vertex has a loop) the complexity of MinHOM(H) is well understood. In this paper, we obtain a full dichotomy for MinHOM(H) when H is an oriented cycle with some loops. Furthermore, we show that this dichotomy is a remarkable progress toward a dichotomy for oriented graphs with some loops.

Original language | English |
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Title of host publication | Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009 |

Volume | 94 |

Publication status | Published - 2009 |

Externally published | Yes |

Event | Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009 - Wellington, New Zealand Duration: 20 Jan 2009 → 23 Jan 2009 |

### Other

Other | Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009 |
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Country | New Zealand |

City | Wellington |

Period | 20/1/09 → 23/1/09 |

### Fingerprint

### Keywords

- Dichotomy
- Homomorphism
- Minimum cost homomorphism
- NP-hardness
- Oriented cycles

### ASJC Scopus subject areas

- Computer Networks and Communications
- Computer Science Applications
- Hardware and Architecture
- Information Systems
- Software

### Cite this

*Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009*(Vol. 94)

**Minimum cost homomorphisms to oriented cycles with some loops.** / Karimi, Mehdi; Gupta, Arvind.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009.*vol. 94, Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009, Wellington, New Zealand, 20/1/09.

}

TY - GEN

T1 - Minimum cost homomorphisms to oriented cycles with some loops

AU - Karimi, Mehdi

AU - Gupta, Arvind

PY - 2009

Y1 - 2009

N2 - For digraphs D and H, a homomorphism of D to H is a mapping f V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) 2 A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i 2 V (H), are nonnegative real costs. The cost of the homomorphism f of D to H is Σ u∈V (D) cf(u)(u). The minimum cost homomorphism for a fixed digraph H, denoted by MinHOM(H), asks whether or not an input digraph D, with nonnegative real costs ci(u), u 2 V (D), i 2 V (H), admits a homomorphism f to H and if it admits one, find a homomorphism of minimum cost. The minimum cost homomorphism problem seems to offer a natural and practical way to model many optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. Our interest is in proving a dichotomy for minimum cost homomorphism problem: we would like to prove that for each digraph H, MinHOM(H) is polynomial-time solvable, or NP-hard. We say that H is a digraph with some loops, if H has at least one loop. For reflexive digraphs H (every vertex has a loop) the complexity of MinHOM(H) is well understood. In this paper, we obtain a full dichotomy for MinHOM(H) when H is an oriented cycle with some loops. Furthermore, we show that this dichotomy is a remarkable progress toward a dichotomy for oriented graphs with some loops.

AB - For digraphs D and H, a homomorphism of D to H is a mapping f V (D)→V (H) such that uv ∈ A(D) implies f(u)f(v) 2 A(H). Suppose D and H are two digraphs, and ci(u), u ∈ V (D), i 2 V (H), are nonnegative real costs. The cost of the homomorphism f of D to H is Σ u∈V (D) cf(u)(u). The minimum cost homomorphism for a fixed digraph H, denoted by MinHOM(H), asks whether or not an input digraph D, with nonnegative real costs ci(u), u 2 V (D), i 2 V (H), admits a homomorphism f to H and if it admits one, find a homomorphism of minimum cost. The minimum cost homomorphism problem seems to offer a natural and practical way to model many optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. Our interest is in proving a dichotomy for minimum cost homomorphism problem: we would like to prove that for each digraph H, MinHOM(H) is polynomial-time solvable, or NP-hard. We say that H is a digraph with some loops, if H has at least one loop. For reflexive digraphs H (every vertex has a loop) the complexity of MinHOM(H) is well understood. In this paper, we obtain a full dichotomy for MinHOM(H) when H is an oriented cycle with some loops. Furthermore, we show that this dichotomy is a remarkable progress toward a dichotomy for oriented graphs with some loops.

KW - Dichotomy

KW - Homomorphism

KW - Minimum cost homomorphism

KW - NP-hardness

KW - Oriented cycles

UR - http://www.scopus.com/inward/record.url?scp=84864003571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864003571&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781920682750

VL - 94

BT - Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009

ER -