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.

翻译：在有限域约束满足问题的计算复杂性分类中，一个核心开放问题是如何为具有马尔采夫多态性的CSP设计更优算法；我们甚至尚未明确这类CSP是否属于NC类。与此相关，这些问题的描述复杂性也尚未解决。卡博内尔曾从均匀多项式时间算法角度研究过的一个重要特例，是具有保守马尔采夫多态性的CSP。我们证明：对于任意具有保守马尔采夫多态性的有限结构B，其约束满足问题可通过对称线性Z2-Datalog程序求解，特别地属于奇偶对数空间复杂性类。此前最著名的算法仅能证明其属于P类。在证明过程中，我们建立了保守马尔采夫代数的结构理论，该理论可能具有独立的研究价值。