Minimum cost homomorphisms to oriented cycles with some loops

Mehdi Karimi*, Arvind Gupta

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review


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
Subtitle of host publicationThe Australasian Theory Symposium, CATS 2009
Publication statusPublished - 1 Dec 2009
EventTheory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009 - Wellington, New Zealand
Duration: 20 Jan 200923 Jan 2009

Publication series

NameConferences in Research and Practice in Information Technology Series
ISSN (Print)1445-1336


ConferenceTheory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009
Country/TerritoryNew Zealand


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


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