Generalising tractable VCSPs defined by symmetric tournament pair multimorphisms

We study optimisation problems that can be formulated as valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions taking finite and infinite costs over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. We are interested in \emph{tractable} constraint languages; that is, languages that give rise to VCSP instances solvable in polynomial time. Cohen et al. (AIJ'06) have shown that constraint languages that admit the MJN multimorphism are tractable. Moreover, using a minimisation algorithm for submodular functions, Cohen et al. (TCS'08) have shown that constraint languages that admit an STP (symmetric tournament pair) multimorphism are tractable. We generalise these results by showing that languages admitting the MJN multimorphism on a subdomain and an STP multimorphisms on the complement of the subdomain are tractable. The algorithm is a reduction to the algorithm for languages admitting an STP multimorphism.

[1]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[2]  Martin Grohe.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2007, JACM.

[3]  Rina Dechter,et al.  Constraint Processing , 1995, Lecture Notes in Computer Science.

[4]  Vladimir Kolmogorov,et al.  The complexity of conservative finite-valued CSPs , 2010, ArXiv.

[5]  Martin C. Cooper,et al.  Constraints, Consistency and Closure , 1998, Artif. Intell..

[6]  Phokion G. Kolaitis,et al.  Conjunctive-Query Containment and Constraint Satisfaction , 2000, J. Comput. Syst. Sci..

[7]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[8]  Martin C. Cooper.  An Optimal k-Consistency Algorithm , 1989, Artif. Intell..

[9]  Francesca Rossi,et al.  Semiring-based constraint satisfaction and optimization , 1997, JACM.

[10]  Phokion G. Kolaitis,et al.  Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics , 2002, CP.

[11]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[12]  Martin C. Cooper,et al.  Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms , 2008, Theor. Comput. Sci..

[13]  Rina Dechter,et al.  Network-based heuristics for constraint satisfaction problems , 1988 .

[14]  Martin C. Cooper,et al.  The complexity of soft constraint satisfaction , 2006, Artif. Intell..

[15]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[16]  Jaroslav Nesetril,et al.  Colouring, constraint satisfaction, and complexity , 2008, Comput. Sci. Rev..

[17]  Rustem Takhanov,et al.  A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem , 2007, STACS.

[18]  Thomas Schiex,et al.  Valued Constraint Satisfaction Problems: Hard and Easy Problems , 1995, IJCAI.

[19]  K. A. Baker,et al.  Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems , 1975 .

[20]  Sanjeev Khanna,et al.  Complexity classifications of Boolean constraint satisfaction problems , 2001, SIAM monographs on discrete mathematics and applications.