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A Fast Algorithm for Maltsev Constraints
Victor Lagerkvist
TL;DR
Maltsev CSPs have polynomial-time algorithms, but their known bounds are inefficient and involve a costly Fix-values step. This paper avoids that step by searching smaller projections, achieving O(mn^2) time for fixed finite languages.
Problem
Known algorithms for Maltsev CSPs are polynomial but far from practical, and their representation-update procedures incur costly repeated prefix searches.
Source: §1 · pp. 1–2Method
The algorithm avoids Fix-values and searches projections of prefix classes using moves induced by the Maltsev operation.
Source: §1 · p. 3Results
For fixed finite languages, the algorithm runs in O(mn^2).
Source: §1 · p. 3Takeaways & Limitations
For fixed finite languages, this bound matches Gaussian elimination.
Source: §1 · p. 3Whether the exponent for k-edge terms can reach O(k) remains open.
Source: §1 · p. 3
Abstract
from arXiv · showhide
The constraint satisfaction problem over a set of relations $\Gamma$ (CSP($\Gamma$)) is the computational problem of deciding if a set of constraints admits at least one solution. The classical complexity for finite-domain CSP($\Gamma$) is settled by the CSP dichotomy theorem: it is tractable if $\Gamma$ 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.
1 Introduction
Maltsev CSPs are tractable, but known algorithms are described as inefficient, with repeated prefix-representation work creating a bottleneck. This paper avoids Fix-values by searching projected tuples, obtaining a quadratic-time bound for fixed finite languages.
1 Introduction
- Maltsev CSPs generalize linear equations through a ternary operation preserving each relation, but known algebraic algorithms are polynomial yet far from practical.The paper contrasts this with local-consistency methods whose limits and applications are well understood.Source: §1 · pp. 1–2
- Bulatov and Dalmau’s algorithm uses Fix-values to construct compact representations, while Dyer and Richerby improve the bound to O(mn^4) but retain repeated prefix-class searches.The paper identifies the repeated construction and search over prefix classes as the shared bottleneck.Source: §1 · p. 2
- Rather than speed up Fix-values, the new approach projects each prefix class onto the constraint scope and the coordinate being processed.These projections contain at most |D|^2|S| tuples.Source: §1 · p. 3
- Maltsev-operation moves reach every tuple in a prefix class, allowing breadth-first search over projected tuples while retaining full tuples needed for the representation.The moves preserve the prefix and reach the class from an initial tuple.Source: §1 · p. 3
- For a fixed, finite constraint language, the algorithm runs in O(m n^2), matching Gaussian elimination, as stated for this restricted setting.Source: §1 · p. 3
- The paper leaves open whether the exponent for the broader k-edge-term setting can reach O(k), with Maltsev as the k = 2 case.Several algorithmic steps carry over, and the exponent may plausibly be lowered.Source: §1 · p. 3
2 Preliminaries
The preliminaries fix a finite domain and establish notation for tuples, projections, CSP instances, Maltsev invariance, and representations. These definitions specify the objects and compact witnesses used by the algorithm.
2 Preliminaries
- The domain D is finite, with d ≥ 2 values.Source: §2 · p. 3
2.1 Tuples, relations, and projections
- An index sequence selects tuple coordinates, and projection returns the selected coordinates in that order.The sequence may repeat indices.Source: §2.1 · p. 3
2.2 The constraint satisfaction problem and invariants
- A CSP instance asks whether an assignment’s projections onto every constraint scope belong to their relations; Maltsev CSPs require those relations to be preserved by a Maltsev operation.A Maltsev operation φ satisfies φ(x, x, y) = φ(y, x, x) = y.Source: §2.2 · p. 4
2.3 Signatures and representations
- A signature records witnessed coordinate-and-value triples, and a compact representation stores one witness per signature triple.Its size is at most |Sig(R)| ≤ nd^2; the paper keeps witnessing pairs explicit.Source: §2.3 · p. 4
3 The Algorithm
The algorithm replaces full prefix-class generation with reachability using restricted Maltsev moves, then applies this search to projected constraint scopes while updating compact solution representations.
3.1 Basic notions and properties
- For a fixed prefix, tuples are reachable from one starting tuple using only moves from representation triples at positions l′ ≥ l.These moves preserve the prefix, so the search avoids first computing a representation of the entire prefix class with Fix-values.Source: §3.1 · p. 6
- Constraints with repeated variables can be replaced by equivalent Maltsev-closed relations over their distinct variables, without increasing relation size.The reduced relation is computed in O(k|S|) time.Source: §3.1 · p. 7
3.2 The algorithm
- O(k n^2 d^5 |S|) per constraint: the algorithm searches projections onto (l,Q), avoiding the full Fix(R,p) representations computed by BD.Each coordinate step uses at most d^2|S| projected tuples, each found by a pass over the compact representation and lifted to a full tuple.Source: §3.2 · p. 8
- Next processes each constraint one scope coordinate at a time, using projected breadth-first searches and lifting discovered projections to full witness tuples.The algorithm maintains a compact representation of assignments satisfying the constraints processed so far.Source: §3.2 · pp. 7–9
4 Correctness and Complexity
The section proves that the compact-representation procedures correctly maintain solution sets and establishes a quadratic-in-n runtime for Solve.
4 Correctness and Complexity
- Reach computes the projected closure and supplies a tuple in the relation witnessing each projected element.Source: §4 · pp. 9–10
- Next-one correctly returns empty exactly when the relation intersected with the constraint is empty, and otherwise returns a compact representation of that intersection.Source: §4 · p. 10
- Next preserves this correctness for a constraint by returning empty exactly when the constrained relation is empty, and otherwise producing its compact representation.Source: §4 · p. 11
- Solve maintains a compact representation of assignments satisfying the constraints processed so far; an empty intermediate set certifies unsatisfiability.Source: §4 · p. 11
- Precomputed transitions encode projected Maltsev moves, while Reach stores vertices in a compact array to support constant-time membership tests.Source: §4 · p. 12
- Solve decides satisfiability and returns a compact representation of all solutions in O(d^5 n^2 ∥I∥) time.Source: §4 · p. 12