TY - CHAP
T1 - Minimum cost homomorphisms to oriented cycles with some loops
AU - Karimi, Mehdi
AU - Gupta, Arvind
PY - 2009/12/1
Y1 - 2009/12/1
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
M3 - Conference paper
AN - SCOPUS:84864003571
SN - 9781920682750
T3 - Conferences in Research and Practice in Information Technology Series
BT - Theory of Computing 2009 - Proceedings of the Fifteenth Computing
T2 - Theory of Computing 2009 - 15th Computing: The Australasian Theory Symposium, CATS 2009
Y2 - 20 January 2009 through 23 January 2009
ER -