TY - CHAP

T1 - Minimum cost homomorphisms to reflexive digraphs

AU - Gupta, Arvind

AU - Hell, Pavol

AU - Karimi, Mehdi

AU - Rafiey, Arash

PY - 2008/5/12

Y1 - 2008/5/12

N2 - For a fixed digraph H, the minimum cost homomorphism problem, MinHOM(H), asks whether an input digraph G, with given costs c i (u), u∈ ∈V(G), i∈ ∈V(H), and an integer k, admits a homomorphism to H of total cost not exceeding k. Minimum cost homomorphism problems encompass many well studied optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H. It is known that MinHOM(H) is polynomial if H has a Min-Max ordering. We prove that for any other reflexive digraph H, the problem MinHOM(H) is NP-complete. (This was earlier conjectured by Gutin and Kim.) Apart from undirected graphs, this is the first general class of digraphs for which such a dichotomy has been proved. Our proof involves a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering, and implies a polynomial test for the existence of a Min-Max ordering in a reflexive digraph H.

AB - For a fixed digraph H, the minimum cost homomorphism problem, MinHOM(H), asks whether an input digraph G, with given costs c i (u), u∈ ∈V(G), i∈ ∈V(H), and an integer k, admits a homomorphism to H of total cost not exceeding k. Minimum cost homomorphism problems encompass many well studied optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H. It is known that MinHOM(H) is polynomial if H has a Min-Max ordering. We prove that for any other reflexive digraph H, the problem MinHOM(H) is NP-complete. (This was earlier conjectured by Gutin and Kim.) Apart from undirected graphs, this is the first general class of digraphs for which such a dichotomy has been proved. Our proof involves a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering, and implies a polynomial test for the existence of a Min-Max ordering in a reflexive digraph H.

KW - Dichotomy

KW - Homomorphism

KW - Minimum cost homomorphism

KW - NP-completeness

KW - Polynomial time algorithm

KW - Reflexive digraph

UR - http://www.scopus.com/inward/record.url?scp=43049120284&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-78773-0_16

DO - 10.1007/978-3-540-78773-0_16

M3 - Conference paper

AN - SCOPUS:43049120284

SN - 3540787720

SN - 9783540787723

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 182

EP - 193

BT - LATIN 2008

T2 - 8th Latin American TheoreticalINformatics Symposium, LATIN 2008

Y2 - 7 April 2008 through 11 April 2008

ER -