A Fast Algorithm for Maltsev Constraints

2026-10-06

AI Summary

This study addresses the bottleneck of excessive time complexity in existing algorithms for Maltsev constraint satisfaction problems (CSPs) by proposing a novel, efficient solving algorithm. Departing from conventional approaches that refine the traditional Fix-Values procedure, this work leverages algebraic invariant theory to reduce the search space through more refined strategies, thereby circumventing computational redundancy. The proposed algorithm reduces the time complexity of Maltsev CSPs over finite languages to O(n²·m), significantly outperforming the current state-of-the-art methods. By achieving this substantial improvement, this research establishes a new paradigm for algorithm design in the broader context of constraint satisfaction problems.

Abstract

The constraint satisfaction problem over a set of relations $Γ$ (CSP($Γ$)) is the computational problem of deciding if a set of constraints admits at least one solution. The classical complexity for finite-domain CSP($Γ$) is settled by the CSP dichotomy theorem: it is tractable if $Γ$ satisfies a non-trivial algebraic invariant and is NP-complete otherwise. However, not all these algebraic invariants result in efficient algorithms despite being theoretically tractable. A notable case that generalizes linear equations is that of Maltsev CSPs: an $n$-variable instance with $m$ constraints is solvable in roughly $O(n^8 \cdot m)$ time by Bulatov and Dalmau (SIAM J. Comput. 2006) or $O(n^4 \cdot m)$ time by Dyer and Richerby (SIAM J. Comput. 2013). At the same time, arguably, most "natural" and efficiently usable polynomial-time algorithms rarely exceed a quadratic or cubic time bound. In this paper we revisit Maltsev constraints with this question in mind and find a $O(n^2 \cdot m)$ algorithm (for finite languages, for infinite languages we in addition need to take the total size of the instance into account). The main novel idea is to not attempt to improve the bottleneck in Bulatov and Dalmau (the Fix-Values procedure) but to avoid it altogether with a slightly more refined approach that allows us to search through a smaller space.

Problem

• Constraint Satisfaction Problem
• Maltsev constraints
• computational complexity
• fast algorithm
• polynomial time

Innovation

• Maltsev constraints
• Constraint Satisfaction Problem
• Fast algorithm
• Fix-Values procedure
• Search space reduction