Tractability in constraint satisfaction problems: a survey

Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP.

[1]  Benoît Larose,et al.  Taylor Terms, Constraint Satisfaction and the Complexity of Polynomial Equations over Finite Algebras , 2006, Int. J. Algebra Comput..

[2]  Christer Bäckström,et al.  A Unifying Approach to Temporal Constraint Reasoning , 1998, Artif. Intell..

[3]  Pedro Barahona,et al.  PSICO: Solving Protein Structures with Constraint Programming and Optimization , 2002, Constraints.

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

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

[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]  Toby Walsh,et al.  Handbook of Constraint Programming , 2006, Handbook of Constraint Programming.

[8]  Peter Jonsson,et al.  A Complete Classification of Tractability in RCC-5 , 1997, J. Artif. Intell. Res..

[9]  Stephan Kreutzer,et al.  Deciding first-order properties of nowhere dense graphs , 2013, STOC.

[10]  Martin C. Cooper,et al.  Hybrid tractability of valued constraint problems , 2010, Artif. Intell..

[11]  Justyna Petke,et al.  The Order Encoding: From Tractable CSP to Tractable SAT , 2011, SAT.

[12]  Martin Grohe.  Generalized Model-Checking Problems for First-Order Logic , 2001, STACS.

[13]  Peter Jeavons,et al.  Building tractable disjunctive constraints , 2000, J. ACM.

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

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

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

[17]  Peter Jeavons,et al.  The Complexity of Constraint Languages , 2006, Handbook of Constraint Programming.

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

[19]  Vladimir Kolmogorov,et al.  The complexity of conservative valued CSPs , 2011, JACM.

[20]  Martin C. Cooper.  Linear-Time Algorithms for Testing the Realisability of Line Drawings of Curved Objects , 1999, Artif. Intell..

[21]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[22]  Martin C. Cooper,et al.  Soft arc consistency revisited , 2010, Artif. Intell..

[23]  Michael R. Fellows,et al.  An O(2O(k)n3) FPT Algorithm for the Undirected Feedback Vertex Set Problem , 2005, COCOON.

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

[25]  Stefan Arnborg,et al.  Efficient Algorithms for Combinatorial Problems with Bounded Decomposability - A Survey. , 1985 .

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

[27]  Marcin Kozik.  A finite set of functions with an EXPTIME-complete composition problem , 2008, Theor. Comput. Sci..

[28]  Paul Walker,et al.  Engineering Dynamic Scheduler for Work Manager , 1998 .

[29]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[30]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[31]  Florian Richoux,et al.  On the Computational Complexity of Monotone Constraint Satisfaction Problems , 2009, WALCOM.

[32]  Manuel Bodirsky,et al.  Relatively quantified constraint satisfaction , 2009, Constraints.

[33]  Peter Jeavons,et al.  Reasoning about temporal relations: The tractable subalgebras of Allen's interval algebra , 2003, JACM.

[34]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness IV: On Completeness for W[P] and PSPACE Analogues , 1995, Ann. Pure Appl. Log..

[35]  Gustav Nordh,et al.  Complexity of SAT Problems, Clone Theory and the Exponential Time Hypothesis , 2013, SODA.

[36]  Eugene C. Freuder.  Complexity of K-Tree Structured Constraint Satisfaction Problems , 1990, AAAI.

[37]  Martin C. Cooper.  Beyond Consistency and Substitutability , 2014, CP.

[38]  Cédric Pralet,et al.  Time-dependent Simple Temporal Networks: Properties and Algorithms , 2012, RAIRO Oper. Res..

[39]  Thomas Schiex,et al.  Semiring-Based CSPs and Valued CSPs: Frameworks, Properties, and Comparison , 1999, Constraints.

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

[41]  Peter van Beek,et al.  On the minimality and global consistency of row-convex constraint networks , 1995, JACM.

[42]  Martin C. Cooper,et al.  Tractable Triangles and Cross-Free Convexity in Discrete Optimisation , 2012, J. Artif. Intell. Res..

[43]  Dániel Marx,et al.  Parameterized complexity of constraint satisfaction problems , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

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

[45]  Michael R. Fellows,et al.  An O(2O(k)n3) FPT Algorithm for the Undirected Feedback Vertex Set Problem , 2005, Theory of Computing Systems.

[46]  Martin Grohe.  The Structure of Tractable Constraint Satisfaction Problems , 2006, MFCS.

[47]  Stanislav Zivny,et al.  The Power of Linear Programming for Valued CSPs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[48]  Martin C. Cooper,et al.  Tractable Constraints on Ordered Domains , 1995, Artif. Intell..

[49]  Marc Gyssens,et al.  A Unifying Framework for Tractable Constraints , 1995, CP.

[50]  Philippe Jégou,et al.  Different Classes of Graphs to Represent Microstructures for CSPs , 2013, GKR.

[51]  Georg Gottlob,et al.  Hypertree decompositions and tractable queries , 1998, PODS '99.

[52]  Martin E. Dyer,et al.  The Complexity of Weighted Boolean #CSP with Mixed Signs , 2009, Theor. Comput. Sci..

[53]  Rina Dechter,et al.  From Local to Global Consistency , 1990, Artif. Intell..

[54]  Yong Gao,et al.  Phase Transition of Tractability in Constraint Satisfaction and Bayesian Network Inference , 2002, UAI.

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

[56]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[57]  A. M. Murray.  The strong perfect graph theorem , 2019, 100 Years of Math Milestones.

[58]  Peter Jeavons,et al.  Constraint Satisfaction Problems on Intervals and Length , 2004, SIAM J. Discret. Math..

[59]  Stanislav Zivny,et al.  The complexity of finite-valued CSPs , 2013, STOC '13.

[60]  Tomás Werner,et al.  A Linear Programming Approach to Max-Sum Problem: A Review , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[61]  Carmen Gervet,et al.  Certainty closure: Reliable constraint reasoning with incomplete or erroneous data , 2006, TOCL.

[62]  Bart Selman,et al.  Backdoors To Typical Case Complexity , 2003, IJCAI.

[63]  Michael R. Fellows,et al.  FIXED-PARAMETER TRACTABILITY AND COMPLETENESS , 2022 .

[64]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[65]  Hubie Chen,et al.  A rendezvous of logic, complexity, and algebra , 2009, CSUR.

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

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

[68]  Hubie Chen,et al.  Arc consistency and friends , 2011, J. Log. Comput..

[69]  Tomasz Łuczak,et al.  A Probabilistic Approach to the Dichotomy Problem , 2006 .

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

[71]  Stefan Szeider,et al.  On the Subexponential Time Complexity of CSP , 2013, AAAI.

[72]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[73]  Peter Jeavons,et al.  The Complexity of Valued Constraint Satisfaction , 2014, Bull. EATCS.

[74]  Manuel Bodirsky,et al.  Maximal infinite-valued constraint languages , 2009, Theor. Comput. Sci..

[75]  Simon de Givry,et al.  Exploiting Tree Decomposition and Soft Local Consistency In Weighted CSP , 2006, AAAI.

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

[77]  Toby Walsh,et al.  Detecting and Exploiting Subproblem Tractability , 2013, IJCAI.

[78]  Pascal Van Hentenryck,et al.  Constraint Satisfaction over Connected Row Convex Constraints , 1997, Artif. Intell..

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

[80]  Manuel Bodirsky,et al.  The complexity of temporal constraint satisfaction problems , 2010, JACM.

[81]  Alfonso Gerevini,et al.  On Finding a Solution in Temporal Constraint Satisfaction Problems , 1997, IJCAI.

[82]  Dániel Marx,et al.  Constraint Satisfaction Parameterized by Solution Size , 2011, ICALP.

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

[84]  Bingkai Lin,et al.  The Parameterized Complexity of k-Biclique , 2014, SODA.

[85]  Michael R. Fellows,et al.  Parameterized complexity: A framework for systematically confronting computational intractability , 1997, Contemporary Trends in Discrete Mathematics.

[86]  Martin E. Dyer,et al.  An Effective Dichotomy for the Counting Constraint Satisfaction Problem , 2010, SIAM J. Comput..

[87]  D ScottAlexander,et al.  Polynomial constraint satisfaction problems, graph bisection, and the Ising partition function , 2009 .

[88]  Rina Dechter,et al.  Network-Based Heuristics for Constraint-Satisfaction Problems , 1987, Artif. Intell..

[89]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[90]  Peter Jonsson,et al.  Computational Complexity of Temporal Constraint Problems , 2005, Handbook of Temporal Reasoning in Artificial Intelligence.

[91]  Satoru Iwata,et al.  A simple combinatorial algorithm for submodular function minimization , 2009, SODA.

[92]  Nadia Creignou,et al.  On Generating All Solutions of Generalized Satisfiability Problems , 1997, RAIRO Theor. Informatics Appl..

[93]  Martin C. Cooper,et al.  On Broken Triangles , 2014, CP.

[94]  Mihalis Yannakakis,et al.  On the Complexity of Database Queries , 1999, J. Comput. Syst. Sci..

[95]  Peter van Beek,et al.  Constraint tightness and looseness versus local and global consistency , 1997, JACM.

[96]  Minghao Yin,et al.  Hybrid Tractable Classes of Binary Quantified Constraint Satisfaction Problems , 2011, AAAI.

[97]  Satoru Iwata,et al.  A fully combinatorial algorithm for submodular function minimization , 2001, SODA '02.

[98]  Peter Jeavons,et al.  A Survey of Tractable Constraint Satisfaction Problems , 1997 .

[99]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[100]  Hubie Chen,et al.  Constraint satisfaction with succinctly specified relations , 2010, J. Comput. Syst. Sci..

[101]  Alex D. Scott,et al.  Polynomial constraint satisfaction problems, graph bisection, and the Ising partition function , 2006, TALG.

[102]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[103]  Andrei A. Bulatov,et al.  Bounded relational width , 2009 .

[104]  Rina Dechter,et al.  Temporal Constraint Networks , 1989, Artif. Intell..

[105]  Peter Jeavons,et al.  Constraint Tractability Theory And Its Application to the Product Development Process for a Constraint−Based Scheduler , 1999 .

[106]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[107]  Martin E. Dyer,et al.  The expressibility of functions on the boolean domain, with applications to counting CSPs , 2011, JACM.

[108]  Ross Willard.  Testing Expressibility Is Hard , 2010, CP.

[109]  Martin C. Cooper,et al.  On Guaranteeing Polynomially Bounded Search Tree Size , 2011, International Conference on Principles and Practice of Constraint Programming.

[110]  Ge Xia,et al.  Improved upper bounds for vertex cover , 2010, Theor. Comput. Sci..

[111]  Paul D. Seymour,et al.  Recognizing Berge Graphs , 2005, Comb..

[112]  Andrei A. Bulatov,et al.  The complexity of the counting constraint satisfaction problem , 2008, JACM.

[113]  Philippe Jégou.  Decomposition of Domains Based on the Micro-Structure of Finite Constraint-Satisfaction Problems , 1993, AAAI.

[114]  Stefan Szeider,et al.  Backdoors into heterogeneous classes of SAT and CSP , 2017, J. Comput. Syst. Sci..

[115]  Wady Naanaa.  Unifying and extending hybrid tractable classes of CSPs , 2013, J. Exp. Theor. Artif. Intell..

[116]  Manuel Bodirsky,et al.  The Complexity of Equality Constraint Languages , 2006, Theory of Computing Systems.

[117]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[118]  Martin C. Cooper,et al.  Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination , 2010, Artif. Intell..

[119]  Martin C. Cooper,et al.  Monotone Temporal Planning: Tractability, Extensions and Applications , 2014, J. Artif. Intell. Res..

[120]  Marko Samer,et al.  Constraint satisfaction with bounded treewidth revisited , 2006, J. Comput. Syst. Sci..

[121]  Yehoshua Bar-Hillel,et al.  The Intrinsic Computational Difficulty of Functions , 1969 .

[122]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[123]  Dániel Marx,et al.  Approximating fractional hypertree width , 2009, TALG.

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

[125]  Willem Jan van Hoeve,et al.  Global Constraints , 2006, Handbook of Constraint Programming.

[126]  Martin C. Cooper,et al.  An Algebraic Theory of Complexity for Discrete Optimization , 2012, SIAM J. Comput..

[127]  Alexandr Kazda,et al.  CSP for binary conservative relational structures , 2011, 1112.1099.

[128]  RobertsonNeil,et al.  Graph minors. XIII , 1994 .

[129]  Georg Gottlob,et al.  Hypertree width and related hypergraph invariants , 2007, Eur. J. Comb..

[130]  D. Welsh.  Complexity: Knots, Colourings and Counting: Link polynomials and the Tait conjectures , 1993 .

[131]  Martin E. Dyer,et al.  The Complexity of Weighted Boolean #CSP , 2009, SIAM J. Comput..

[132]  Justin Pearson,et al.  Closure Functions and Width 1 Problems , 1999, CP.

[133]  Víctor Dalmau,et al.  Generalized majority-minority operations are tractable , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[134]  Phokion G. Kolaitis,et al.  Closures and dichotomies for quantified constraints , 2006, Electron. Colloquium Comput. Complex..

[135]  David A. Cohen,et al.  Domain permutation reduction for constraint satisfaction problems , 2008, Artif. Intell..

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

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

[138]  L. Barto.  Finitely Related Algebras in Congruence Distributive Varieties Have Near Unanimity Terms , 2013, Canadian Journal of Mathematics.

[139]  Philippe David,et al.  Using Pivot Consistency to Decompose and Solve Functional CSPs , 1994, J. Artif. Intell. Res..

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

[141]  Vladimir Kolmogorov,et al.  The Complexity of General-Valued CSPs , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[142]  Umberto Bertelè,et al.  Nonserial Dynamic Programming , 1972 .

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

[144]  Dániel Marx,et al.  Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries , 2009, JACM.

[145]  Daniël Paulusma,et al.  Satisfiability of acyclic and almost acyclic CNF formulas , 2011, Theor. Comput. Sci..

[146]  Martin C. Cooper,et al.  On Backdoors to Tractable Constraint Languages , 2014, CP.

[147]  Martin C. Cooper,et al.  The tractability of CSP classes defined by forbidden patterns , 2012, J. Artif. Intell. Res..

[148]  Dániel Marx.  Tractable Structures for Constraint Satisfaction with Truth Tables , 2009, Theory of Computing Systems.

[149]  Dániel Marx,et al.  Constraint solving via fractional edge covers , 2006, SODA '06.

[150]  Stanislav Zivny,et al.  The Complexity of Valued Constraint Satisfaction Problems , 2012, Cognitive Technologies.

[151]  Peter Jeavons.  Constructing Constraints , 1998, CP.

[152]  Todd Niven,et al.  On Maltsev Digraphs , 2011, Electron. J. Comb..

[153]  Martin C. Cooper,et al.  A Dichotomy for 2-Constraint Forbidden CSP Patterns , 2012, AAAI.

[154]  Pavol Hell,et al.  Full Constraint Satisfaction Problems , 2006, SIAM J. Comput..

[155]  Thomas Schiex,et al.  DARN! A Weighted Constraint Solver for RNA Motif Localization , 2007, Constraints.

[156]  Jaroslav Nesetril,et al.  A Probabilistic Approach to the Dichotomy Problem , 2006, SIAM J. Comput..

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

[158]  Peter Jeavons,et al.  Perfect Constraints Are Tractable , 2008, CP.

[159]  R. Willard,et al.  Characterizations of several Maltsev conditions , 2015 .

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

[161]  Manuel Bodirsky,et al.  Non-dichotomies in Constraint Satisfaction Complexity , 2008, ICALP.

[162]  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).

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

[164]  Dov M. Gabbay,et al.  Handbook of Temporal Reasoning in Artificial Intelligence , 2005, Handbook of Temporal Reasoning in Artificial Intelligence.

[165]  Manolis Koubarakis,et al.  Tractable Disjunctions of Linear Constraints , 1996, CP.

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

[167]  Mihalis Yannakakis,et al.  On the complexity of database queries (extended abstract) , 1997, PODS.

[168]  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..

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