AI 中文总结
本文针对 Maltsev 约束满足问题，提出一种 O(n^2·m) 的快速算法，通过避免 Fix-Values 瓶颈并缩小搜索空间，显著优于现有 O(n^4·m) 算法。
AI 中文摘要
约束满足问题（CSP）是在一组关系 Γ 上定义的（CSP(Γ)），其计算任务是判定一组约束是否至少存在一个解。有限域 CSP(Γ) 的经典复杂性由 CSP 二分定理解决：若 Γ 满足非平凡代数不变量，则该问题可解；否则为 NP 完全。然而，并非所有这些代数不变量都能产生高效算法，尽管它们在理论上可解。一个推广线性方程的重要情形是 Maltsev CSP：具有 m 个约束的 n 变量实例可由 Bulatov 和 Dalmau（SIAM J. Comput. 2006）在约 O(n^8 · m) 时间内求解，或由 Dyer 和 Richerby（SIAM J. Comput. 2013）在 O(n^4 · m) 时间内求解。同时，可以说，大多数“自然”且可高效使用的多项式时间算法很少超过二次或三次时间界。在本文中，我们带着这一问题重新审视 Maltsev 约束，并找到一个 O(n^2 · m) 算法（对于有限语言；对于无限语言，我们还需考虑实例的总大小）。主要的新颖思想是：不试图改进 Bulatov 和 Dalmau 中的瓶颈（Fix-Values 过程），而是通过一种略微更精细的方法完全避开它，从而允许我们在更小的空间中搜索。
英文摘要
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.
