TY - GEN

T1 - A dichotomy theorem for the general minimum cost homomorphism problem

AU - Takhanov, Rustem

PY - 2010/12/1

Y1 - 2010/12/1

N2 - In the constraint satisfaction problem (CSP), the aim is to find an assignment of values to a set of variables subject to specified constraints. In the minimum cost homomorphism problem (MinHom), one is additionally given weights cva for every variable v and value a, and the aim is to find an assignment f to the variables that minimizes Σv c vf(v). Let MinHom(Γ) denote the MinHom problem parameterized by the set of predicates allowed for constraints. MinHom (Γ) is related to many well-studied combinatorial optimization problems, and concrete applications can be found in, for instance, defence logistics and machine learning. We show that MinHom(Γ) can be studied by using algebraic methods similar to those used for CSPs. With the aid of algebraic techniques, we classify the computational complexity of MinHom (Γ) for all choices of Γ. Our result settles a general dichotomy conjecture previously resolved only for certain classes of directed graphs, [Gutin, Hell, Rafiey, Yeo, European J. of Combinatorics, 2008].

AB - In the constraint satisfaction problem (CSP), the aim is to find an assignment of values to a set of variables subject to specified constraints. In the minimum cost homomorphism problem (MinHom), one is additionally given weights cva for every variable v and value a, and the aim is to find an assignment f to the variables that minimizes Σv c vf(v). Let MinHom(Γ) denote the MinHom problem parameterized by the set of predicates allowed for constraints. MinHom (Γ) is related to many well-studied combinatorial optimization problems, and concrete applications can be found in, for instance, defence logistics and machine learning. We show that MinHom(Γ) can be studied by using algebraic methods similar to those used for CSPs. With the aid of algebraic techniques, we classify the computational complexity of MinHom (Γ) for all choices of Γ. Our result settles a general dichotomy conjecture previously resolved only for certain classes of directed graphs, [Gutin, Hell, Rafiey, Yeo, European J. of Combinatorics, 2008].

KW - Constraint satisfaction problem

KW - Minimum cost homomorphisms problem

KW - Perfect graphs

KW - Relational clones

KW - Supervised learning

UR - http://www.scopus.com/inward/record.url?scp=84880277745&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880277745&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2010.2493

DO - 10.4230/LIPIcs.STACS.2010.2493

M3 - Conference contribution

AN - SCOPUS:84880277745

SN - 9783939897163

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 657

EP - 668

BT - STACS 2010 - 27th International Symposium on Theoretical Aspects of Computer Science

T2 - 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010

Y2 - 4 March 2010 through 6 March 2010

ER -