Summary
The paper shows a new way to solve a special kind of puzzle called Maltsev constraint satisfaction problems using computer algorithms that are very efficient in space and counting. This new approach does not require directly knowing the special rules (Maltsev polymorphisms) of the puzzle but uses a graph-based method relying on smaller building blocks within the puzzle. The authors explain that this connects these problems to a known complexity class tied to calculating determinants in linear algebra. This work places these puzzles in a mathematically well-understood framework related to how computers process information with limited memory and counting ability.
What this means in practice
- •For complexity engineers: Design algorithms for constraint problems that fit in efficient deterministic logspace classes with counting capabilities.
- •For database query optimizers: Optimize query evaluation plans by identifying structural properties related to Maltsev polymorphisms to reduce computational resources.
A theory result. No direct application yet.
Abstract
In this article, we prove that the problem of solving $\operatorname{CSP}(\mathbf{A})$, where $\mathbf{A}$ is a finite relational template which admits a Maltsev polymorphism is in a specific complexity class DET, which is related to the complexity of computing the determinant of a matrix with integer entries. Such a class is intimately related to well-studied MOD-logspace classes in the theory of computational complexity. To prove this fact, we develop a new algorithm for solving syntactically simple binary instances of Maltsev constraint satisfaction problems, rather different from the well-known Bulatov-Dalmau algorithm, which does not require the explicit use or knowledge of a Maltsev polymorphism of the template but, rather, utilizes a graph whose vertices are 2-generated subuniverses of $\mathbb{A}$, where $\mathbb{A}$ is the Maltsev algebra parametrizing $\operatorname{CSP}(\mathbf{A})$. The theoretical importance of this algorithm is reflected in two facts: (1) it places the problem $ \operatorname{CSP}(\mathbf{A})$ in a complexity class related to the deterministic logspace with counting, which, in itself, has a strong connection to a variety of standard algorithmic problems in linear algebra, and (2) it only makes use of the relational structure of the template without the need for the explicit use of a compatible Maltsev polymorphism, depending entirely on the strong ``symmetry" of constraints compatible with such polymorphisms and the knowledge of 2-generated subuniverses of $\mathbb{A}$.