The Complexity of Boolean Constraint Isomorphism

We consider the Boolean constraint isomorphism problem, that is, the problem of determining whether two sets of Boolean constraint applications can be made equivalent by renaming the variables. We show that depending on the set of allowed constraints, the problem is either coNP-hard and GI-hard, equivalent to graph isomorphism, or polynomial-time solvable. This establishes a complete classification of the complexity of the problem, and moreover, it identifies exactly all those cases in which Boolean constraint isomorphism is polynomial-time many-one equivalent to graph isomorphism, the best-known and best-examined isomorphism problem in theoretical computer science.

[1]  Thomas Thierauf,et al.  The Computational Complexity of Equivalence and Isomorphism Problems , 2000, Lecture Notes in Computer Science.

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

[3]  Nadia Creignou,et al.  A Dichotomy Theorem for Maximum Generalized Satisfiability Problems , 1995, J. Comput. Syst. Sci..

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

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

[6]  Arnaud Durand,et al.  The Inference Problem for Propositional Circumscription of Affine Formulas Is coNP-Complete , 2003, STACS.

[7]  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..

[8]  Heribert Vollmer,et al.  Optimal satisfiability for propositional calculi and constraint satisfaction problems , 2003, Inf. Comput..

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

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

[11]  Manindra Agrawal,et al.  The Formula Isomorphism Problem , 2000, SIAM J. Comput..

[12]  Alfred Horn,et al.  On sentences which are true of direct unions of algebras , 1951, Journal of Symbolic Logic.

[13]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[14]  Uri Zwick,et al.  Finding almost-satisfying assignments , 1998, STOC '98.

[15]  Heribert Vollmer,et al.  Equivalence and Isomorphism for Boolean Constraint Satisfaction , 2002, CSL.

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

[17]  Laurent Juban,et al.  Dichotomy Theorem for the Generalized Unique Satisfiability Problem , 1999, FCT.

[18]  Marc Gyssens,et al.  How to Determine the Expressive Power of Constraints , 1999, Constraints.

[19]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

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

[21]  Martha Sideri,et al.  The Inverse Satisfiability Problem , 1996, SIAM J. Comput..

[22]  Peter Jeavons,et al.  Constraint Satisfaction Problems and Finite Algebras , 2000, ICALP.

[23]  Desh Ranjan,et al.  On the Computational Complexity of Some Classical Equivalence Relations on Boolean Functions , 1998, Theory of Computing Systems.

[24]  Phokion G. Kolaitis,et al.  The complexity of minimal satisfiability problems , 2001, Inf. Comput..

[25]  Vikraman Arvind,et al.  Graph isomorphism is in SPP , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[26]  Luca Trevisan,et al.  The Approximability of Constraint Satisfaction Problems , 2001, SIAM J. Comput..

[27]  Phokion G. Kolaitis,et al.  A Dichotomy in the Complexity of Propositional Circumscription , 2001, Theory of Computing Systems.

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

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

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

[31]  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..

[32]  Peter Jeavons,et al.  Quantified Constraints: Algorithms and Complexity , 2003, CSL.

[33]  Jacobo Torán,et al.  On the hardness of graph isomorphism , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[34]  Gustav Nordh,et al.  A Trichotomy in the Complexity of Propositional Circumscription , 2005, LPAR.