Characterising the complexity of constraint satisfaction problems defined by 2-constraint forbidden patterns

Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of binary CSP instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover several new tractable classes including, for example, a novel generalisation of 2SAT. We then extend this dichotomy to existential patterns which are only forbidden on specific domain values.

[1]  Martin C. Cooper,et al.  Hierarchically Nested Convex VCSP , 2011, CP.

[2]  Dániel Marx.  Can you beat treewidth? , 2007, FOCS.

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

[4]  Martin Grohe,et al.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

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

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

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

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

[9]  Martin C. Cooper.  Fundamental Properties of Neighbourhood Substitution in Constraint Satisfaction Problems , 1997, Artif. Intell..

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

[11]  Eugene C. Freuder.  Eliminating Interchangeable Values in Constraint Satisfaction Problems , 1991, AAAI.

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

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

[14]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

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

[16]  Dániel Marx,et al.  Can you beat treewidth? , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[17]  Philippe Jégou,et al.  Microstructures for CSPs with Constraints of Arbitrary Arity , 2013, SARA.

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

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

[20]  Roland H. C. Yap,et al.  An optimal coarse-grained arc consistency algorithm , 2005, Artif. Intell..

[21]  Martin C. Cooper,et al.  Variable Elimination in Binary CSP via Forbidden Patterns , 2013, IJCAI.

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

[23]  Philippe Jégou,et al.  A hybrid tractable class for non-binary CSPs , 2013, 2013 IEEE 25th International Conference on Tools with Artificial Intelligence.

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