Finitely Tractable Promise Constraint Satisfaction Problems

The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].

[1]  Libor Barto,et al.  Promises Make Finite (Constraint Satisfaction) Problems Infinitary , 2019, 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[2]  Manuel Bodirsky.  Constraint Satisfaction Problems with Infinite Templates , 2008, Complexity of Constraints.

[3]  Guofeng Deng,et al.  Sandwiches for promise constraint satisfaction , 2020, Algebra universalis.

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

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

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

[7]  Venkatesan Guruswami,et al.  Promise Constraint Satisfaction: Structure Theory and a Symmetric Boolean Dichotomy , 2018, SODA.

[8]  Libor Barto,et al.  The wonderland of reflections , 2015, Israel Journal of Mathematics.

[9]  Dmitriy Zhuk,et al.  A Proof of CSP Dichotomy Conjecture , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

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

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

[12]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[13]  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..

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

[15]  Marcin Kozik,et al.  Dichotomy for symmetric Boolean PCSPs , 2019, ICALP.

[16]  Jakub Bulín,et al.  Algebraic approach to promise constraint satisfaction , 2018, STOC.

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