A Dichotomy Theorem for Nonuniform CSPs

In a non-uniform Constraint Satisfaction problem CSP(Γ), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from Γ. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language \Gm the problem CSP(Γ) is either solvable in polynomial time or is NP-complete. It was proposed by Feder and Vardi in their seminal 1993 paper. In this paper we confirm the Dichotomy Conjecture.

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

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

[3]  R. McKenzie,et al.  Commutator theory for congruence modular varieties , 1989 .

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

[5]  Tomás Feder,et al.  Monotone monadic SNP and constraint satisfaction , 1993, STOC.

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

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

[8]  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..

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

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

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

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

[13]  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..

[14]  G. McNulty,et al.  Algebras, Lattices, Varieties , 1988, Mathematical Surveys and Monographs.

[15]  Andrei A. Bulatov,et al.  Tractable conservative constraint satisfaction problems , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

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

[17]  A. Bulatov.  A graph of a relational structure and constraint satisfaction problems , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[18]  Neil Immerman,et al.  The complexity of satisfiability problems: Refining Schaefer's theorem , 2009, J. Comput. Syst. Sci..

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

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

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

[22]  Claude Tardif,et al.  A Characterisation of First-Order Constraint Satisfaction Problems , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[23]  Pawel M. Idziak,et al.  Tractability and learnability arising from algebras with few subpowers , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[24]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[25]  Andrei A. Bulatov,et al.  Dualities for Constraint Satisfaction Problems , 2008, Complexity of Constraints.

[26]  M. Maróti,et al.  Existence theorems for weakly symmetric operations , 2008 .

[27]  Heribert Vollmer,et al.  Complexity of Constraints - An Overview of Current Research Themes [Result of a Dagstuhl Seminar] , 2008, Complexity of Constraints.

[28]  Andrei A. Bulatov,et al.  Recent Results on the Algebraic Approach to the CSP , 2008, Complexity of Constraints.

[29]  Andrei A. Krokhin,et al.  Majority constraints have bounded pathwidth duality , 2008, Eur. J. Comb..

[30]  Libor Barto,et al.  The CSP Dichotomy Holds for Digraphs with No Sources and No Sinks (A Positive Answer to a Conjecture of Bang-Jensen and Hell) , 2008, SIAM J. Comput..

[31]  R. McKenzie,et al.  Varieties with few subalgebras of powers , 2009 .

[32]  Ralph Freese,et al.  On the Complexity of Some Maltsev Conditions , 2009, Int. J. Algebra Comput..

[33]  Mario Szegedy,et al.  A new line of attack on the dichotomy conjecture , 2009, STOC '09.

[34]  Andrei A. Bulatov,et al.  Complexity of conservative constraint satisfaction problems , 2011, TOCL.

[35]  Andrei A. Bulatov,et al.  On the CSP Dichotomy Conjecture , 2011, CSR.

[36]  Libor Barto,et al.  The Dichotomy for Conservative Constraint Satisfaction Problems Revisited , 2011, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[37]  Libor Barto,et al.  Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem , 2012, Log. Methods Comput. Sci..

[38]  Libor Barto,et al.  Near Unanimity Constraints Have Bounded Pathwidth Duality , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[39]  Pushmeet Kohli,et al.  Tractability: Practical Approaches to Hard Problems , 2013 .

[40]  Dmitriy Zhuk.  On Key Relations Preserved by a Weak Near-Unanimity Function , 2014, 2014 IEEE 44th International Symposium on Multiple-Valued Logic.

[41]  Georg Gottlob,et al.  Treewidth and Hypertree Width , 2014, Tractability.

[42]  Libor Barto,et al.  Constraint Satisfaction Problems Solvable by Local Consistency Methods , 2014, JACM.

[43]  Libor Barto,et al.  THE CONSTRAINT SATISFACTION PROBLEM AND UNIVERSAL ALGEBRA , 2015, The Bulletin of Symbolic Logic.

[44]  Andrei A. Bulatov.  Graphs of relational structures: restricted types , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[45]  Andrei A. Bulatov,et al.  Conservative constraint satisfaction re-revisited , 2014, J. Comput. Syst. Sci..

[46]  Prasad Raghavendra,et al.  Correlation Decay and Tractability of CSPs , 2016, ICALP.

[47]  Libor Barto,et al.  The collapse of the bounded width hierarchy , 2016, J. Log. Comput..

[48]  Andrei A. Bulatov.  Graphs of finite algebras, edges, and connectivity , 2016, ArXiv.

[49]  Marcin Kozik.  Weak consistency notions for all the CSPs of bounded width∗ , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[50]  K. Kearnes.  Idempotent Simple Algebras , 2017 .

[51]  Hubie Chen,et al.  Asking the Metaquestions in Constraint Tractability , 2016, TOCT.

[52]  Libor Barto,et al.  Polymorphisms, and How to Use Them , 2017, The Constraint Satisfaction Problem.

[53]  Andrei A. Bulatov.  Constraint satisfaction problems over semilattice block Mal'tsev algebras , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[54]  Andrei A. Krokhin,et al.  The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231) , 2018, Dagstuhl Reports.

[55]  Constraint Satisfaction Problems over semilattice block Mal'tsev algebras , 2019, Inf. Comput..