Conservative Maltsev Constraint Satisfaction Problems

TL;DR

This paper proves that CSPs with conservative Maltsev polymorphisms can be solved by symmetric linear Z2-Datalog programs, placing them in the complexity class parity-L, and develops a new structure theory for these algebras.

Contribution

It establishes a new complexity classification for CSPs with conservative Maltsev polymorphisms and introduces a structure theory for conservative Maltsev algebras.

Findings

CSPs with conservative Maltsev polymorphisms are in parity-L.

Develops a structure theory for conservative Maltsev algebras.

Shows CSPs with these polymorphisms can be solved by symmetric linear Z2-Datalog.

Abstract

One of the central open problems to classify the computational complexity of finite-domain constraint satisfaction problems within P is to prove better algorithmic results for CSPs with a Maltsev polymorphism; we do not even know whether these CSPs are in NC. Relatedly, the descriptive complexity of these problems is open as well. An important special case, previously studied by Carbonell from the perspective of uniform polynomial time-algorithms, are CSPs with a conservative Maltsev polymorphism. We show that for every finite structure B with a conservative Maltsev polymorphism, the CSP for B can be solved by a symmetric linear Z2-Datalog program, and in particular is in the complexity class parity-L. Previously, the best known algorithms just showed containment in P. In our proof we develop a structure theory for conservative Maltsev algebras which might be of independent interest.