Complexity of conservative constraint satisfaction problems

In a constraint satisfaction problem (CSP), the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NP-complete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in polynomial time. Such restrictions are usually imposed by specifying a constraint language, that is, a set of relations that are allowed to be used as constraints. A principal research direction aims to distinguish those constraint languages that give rise to tractable CSPs from those that do not. We achieve this goal for the important version of the CSP, in which the set of values for each individual variable can be restricted arbitrarily. Restrictions of this type can be studied by considering those constraint languages which contain all possible unary constraints; we call such languages conservative. We completely characterize conservative constraint languages that give rise to polynomial time solvable CSP classes. In particular, this result allows us to obtain a complete description of those (directed) graphs H for which the List H-Coloring problem is solvable in polynomial time. The result, the solving algorithm, and the proofs heavily use the algebraic approach to CSP developed in Jeavons et al. [1997], Jeavons [1998], Bulatov et al. [2005], and Bulatov and Jeavons [2001b, 2003].

[1]  Peter Jeavons,et al.  Tractable constraints closed under a binary operation , 2000 .

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

[3]  Marc Gyssens,et al.  Closure properties of constraints , 1997, JACM.

[4]  P. Jeavons.  Algebraic structures in combinatorial problems , 2001 .

[5]  Zsolt Tuza,et al.  Algorithmic complexity of list colorings , 1994, Discret. Appl. Math..

[6]  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..

[7]  Klaus Denecke,et al.  Universal Algebra and Applications in Theoretical Computer Science , 2018 .

[8]  A BulatovAndrei.  Complexity of conservative constraint satisfaction problems , 2011 .

[9]  Roman Barták,et al.  Constraint Processing , 2009, Encyclopedia of Artificial Intelligence.

[10]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[11]  Pavol Hell,et al.  List Homomorphisms to Reflexive Graphs , 1998, J. Comb. Theory, Ser. B.

[12]  Peter Jeavons,et al.  An Algebraic Approach to Multi-sorted Constraints , 2003, CP.

[13]  Jaroslav Nesetril,et al.  The complexity of H-colouring of bounded degree graphs , 2000, Discret. Math..

[14]  Sulamita Klein,et al.  List Partitions , 2003, SIAM J. Discret. Math..

[15]  Víctor Dalmau,et al.  A new tractable class of constraint satisfaction problems , 2005, Annals of Mathematics and Artificial Intelligence.

[16]  Pavol Hell,et al.  List Homomorphisms and Circular Arc Graphs , 1999, Comb..

[17]  Peter Jeavons,et al.  The complexity of maximal constraint languages , 2001, STOC '01.

[18]  Justin Pearson,et al.  Constraints and universal algebra , 1998, Annals of Mathematics and Artificial Intelligence.

[19]  Emil L. Post.  The two-valued iterative systems of mathematical logic , 1942 .

[20]  P. Hell,et al.  Sparse pseudo-random graphs are Hamiltonian , 2003 .

[21]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraints on a three-element set , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[22]  Ugo Montanari,et al.  Networks of constraints: Fundamental properties and applications to picture processing , 1974, Inf. Sci..

[23]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

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

[25]  Andrei A. Bulatov,et al.  Mal'tsev constraints are tractable , 2002, Electron. Colloquium Comput. Complex..

[26]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[27]  Martin E. Dyer,et al.  The complexity of counting graph homomorphisms , 2000, Random Struct. Algorithms.

[28]  Nicholas Pippenger,et al.  Theories of computability , 1997 .

[29]  Phokion G. Kolaitis,et al.  A Game-Theoretic Approach to Constraint Satisfaction , 2000, AAAI/IAAI.

[30]  Martin C. Cooper,et al.  Characterising Tractable Constraints , 1994, Artif. Intell..

[31]  A. Bulatov.  Combinatorial problems raised from 2-semilattices , 2006 .

[32]  Andrei A. Bulatov,et al.  A Simple Algorithm for Mal'tsev Constraints , 2006, SIAM J. Comput..

[33]  Peter Jeavons,et al.  Functions of multiple-valued logic and the complexity of constraint satisfaction: A short survey , 2003, 33rd International Symposium on Multiple-Valued Logic, 2003. Proceedings..

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

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

[36]  B. Larose,et al.  Bounded width problems and algebras , 2007 .

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

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

[39]  Ágnes Szendrei,et al.  Clones in universal algebra , 1986 .

[40]  Phokion G. Kolaitis.  Constraint Satisfaction, Databases, and Logic , 2003, IJCAI.

[41]  Richard C. Brewster,et al.  Near-Unanimity Functions and Varieties of Reflexive Graphs , 2008, SIAM J. Discret. Math..

[42]  Georg Gottlob,et al.  Hypertree decompositions and tractable queries , 1998, J. Comput. Syst. Sci..

[43]  Reinhard Pöschel,et al.  Funktionen- und Relationenalgebren , 1979 .

[44]  Víctor Dalmau,et al.  Constraint Satisfaction Problems in Non-deterministic Logarithmic Space , 2002, ICALP.