Minimum cost homomorphisms to oriented cycles with some loops

Mehdi Karimi, Arvind Gupta

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish
Title of host publicationTheory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009
Volume94
Publication statusPublished - 2009
Externally publishedYes
EventTheory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009 - Wellington, New Zealand
Duration: 20 Jan 200923 Jan 2009

Other

OtherTheory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009
CountryNew Zealand
CityWellington
Period20/1/0923/1/09

Fingerprint

Costs
Repair
Polynomials

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

Karimi, M., & Gupta, A. (2009). Minimum cost homomorphisms to oriented cycles with some loops. In 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.

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Karimi, M & Gupta, A 2009, Minimum cost homomorphisms to oriented cycles with some loops. in 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.
Karimi M, Gupta A. Minimum cost homomorphisms to oriented cycles with some loops. In Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009. Vol. 94. 2009
Karimi, Mehdi ; Gupta, Arvind. / Minimum cost homomorphisms to oriented cycles with some loops. Theory of Computing 2009 - Proceedings of the Fifteenth Computing: The Australasian Theory Symposium, CATS 2009. Vol. 94 2009.
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