The Complexity of Equality Constraint Languages

We apply the algebraic approach to infinite-valued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if all the constraint types are Boolean combinations of the equality relation, and we call the corresponding constraint languages equality constraint languages. We show that an equality constraint language is tractable if it admits a constant unary or an injective binary polymorphism, and is NP-complete otherwise.

[1]  D. Geiger.  CLOSED SYSTEMS OF FUNCTIONS AND PREDICATES , 1968 .

[2]  L. A. Kaluzhnin,et al.  Galois theory for post algebras. I , 1969 .

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

[4]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

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

[6]  P. Cameron,et al.  Oligomorphic permutation groups , 1990 .

[7]  Bernhard Nebel,et al.  Reasoning about temporal relations: a maximal tractable subclass of Allen's interval algebra , 1994, JACM.

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

[9]  Wilfrid Hodges,et al.  A Shorter Model Theory , 1997 .

[10]  Peter van Beek,et al.  Local and Global Relational Consistency , 1995, Theor. Comput. Sci..

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

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

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

[14]  L. Heindorf.  The maximal clones on countable sets that include all permutations , 2002 .

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

[16]  Jaroslav Nesetril,et al.  Constraint Satisfaction with Countable Homogeneous Templates , 2003, CSL.

[17]  M. Pinsker.  Precomplete Clones on Infinite Sets which are Closed under Conjugation , 2004, math/0409217.

[18]  Manuel Bodirsky,et al.  Constraint satisfaction with infinite domains , 2004 .

[19]  M. Pinsker.  Maximal clones on uncountable sets that include all permutations , 2004, math/0401103.

[20]  M. Pinsker.  THE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET , 2004, math/0410406.

[21]  Manuel Bodirsky,et al.  The Core of a Countably Categorical Structure , 2005, STACS.

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

[23]  Manuel Bodirsky,et al.  Datalog and Constraint Satisfaction with Infinite Templates , 2006, STACS.