Dalmau's Algorithm for GMM CSPs

• Dalmau's Algorithm is a polynomial-time method for solving CSPs whose constraints are preserved by generalized majority-minority operations using compact representations.
• It employs a refined signature approach and maintains small projections, combining algebraic properties of Mal'tsev and near-unanimity operations.
• The algorithm's soundness is formalized in V¹, leading to polynomial-size extended Frege proofs for unsatisfiable instances in the CSP framework.

Dalmau's Algorithm is the polynomial-time algorithm, as treated in the proof-complexity study of Mal'tsev and generalized majority-minority constraint satisfaction problems, for solving CSPs whose constraint language is preserved by a generalized majority-minority (GMM) operation. In that treatment, the algorithm is presented as the GMM analogue of the Bulatov–Dalmau algorithm for Mal'tsev CSPs: it works by maintaining compact representations of solution sets under successive constraint addition, and its soundness can be formalized in the bounded-arithmetic theory $V^1$, yielding polynomial-size extended Frege proofs for propositional encodings of unsatisfiable instances (Gaysin, 1 Aug 2025).

1. Problem class and tractability statement

The algorithm applies to instances of $\mathrm{CSP}(A)$ in which the template $A$ admits a GMM polymorphism; equivalently, the relevant relations are invariant under such an operation. In the formulation cited in the proof-complexity paper, the tractability statement is:

$$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$

The source notes a small variable mismatch in that statement, but identifies the intended content as the standard one: CSPs defined by relations invariant under a GMM polymorphism are tractable (Gaysin, 1 Aug 2025).

This situates Dalmau's Algorithm inside the algebraic CSP program that refines the broad dichotomy between NP-complete and polynomial-time finite-domain CSPs. The same source places that context against the CSP dichotomy theorem of Zhuk (2017) and Bulatov (2017), and then treats GMM tractability as a further structural subdivision within the polynomial-time side. A plausible implication is that Dalmau's Algorithm is best understood not as an isolated procedure, but as an algorithmic manifestation of a specific polymorphism condition.

2. GMM operations as a unifying algebraic condition

The defining algebraic idea is that a GMM operation combines two previously central tractability patterns. The proof-complexity account states that a GMM operation is a $(k+1)$-ary operation that, when restricted to any two-element subset $\{a,b\}$, behaves either like a near-unanimity operation or like a minority/Mal'tsev-style operation (Gaysin, 1 Aug 2025).

Operation type	Identity
Mal'tsev	$p(x,y,y)=p(y,y,x)=x$
Near-unanimity	$$p(x,y,\dots,y)=p(y,x,\dots,y)=\cdots=p(y,\dots,y,x)=y$$
GMM	On each $\{a,b\}$, either majority-like or minority-like

The same source explicitly describes GMM operations as a common generalization of near-unanimity and Mal'tsev operations. It further states that the reason for combining them is that in both cases every subalgebra of $A^n$ has a small generating set. That structural fact is the operative mechanism behind both the Mal'tsev algorithm and Dalmau's Algorithm: instead of storing exponentially large relations, the algorithms manipulate compact representations of subalgebras or solution sets.

This also clarifies a recurring misconception. Dalmau's Algorithm is not merely the Mal'tsev algorithm with a different name. The paper describes the two procedures as “very similar,” but the GMM setting requires a refined notion of representation because majority-like and minority-like behavior coexist and must be handled differently.

3. Signatures, projections, and compact representations

The proof-complexity treatment develops Dalmau's Algorithm by first reviewing the Mal'tsev machinery. For a relation $\mathrm{CSP}(A)$0, a pair of tuples $\mathrm{CSP}(A)$1 witnesses $\mathrm{CSP}(A)$2 if the tuples agree on all coordinates $\mathrm{CSP}(A)$3 and differ at coordinate $\mathrm{CSP}(A)$4 by $\mathrm{CSP}(A)$5 versus $\mathrm{CSP}(A)$6. The Mal'tsev signature is then

$\mathrm{CSP}(A)$7

A representation $\mathrm{CSP}(A)$8 is one with the same signature, and a compact representation has size at most $\mathrm{CSP}(A)$9. The key generating theorem is stated as follows:

$A$0

For GMM, the notion of signature is modified so that only minority pairs contribute to the signature component. The paper defines a signature relative to a GMM operation $A$1 as the set of triples $A$2 such that $A$3 is a minority pair and is witnessed by elements of $A$4. A representation relative to GMM must preserve both that GMM signature and all small projections $A$5 for every $A$6 with $A$7 (Gaysin, 1 Aug 2025).

The corresponding generation theorem is:

$A$8

The source describes the proof as “very similar” to the Mal'tsev case, but with induction and a case distinction according to whether each pair $A$9 is majority or minority. This suggests that the core invariant of Dalmau's Algorithm is not merely syntactic compression, but an algebraically calibrated summary of a relation: minority-pair signature data plus all projections up to arity $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$0.

4. Algorithmic organization and principal subroutines

The algorithmic skeleton mirrors the Mal'tsev procedure. One starts with the solution set for the empty instance, $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$1, maintained through a compact representation. Constraints are then added one by one, and after each step the current representation is updated by a subroutine called next. If the final representation is empty, the instance is reported unsatisfiable (Gaysin, 1 Aug 2025).

The proof-complexity paper emphasizes that the GMM version uses the same overall structure as the Mal'tsev algorithm, but the maintained representation must additionally remember small projections. In consequence, the GMM representation contains two kinds of objects: witness maps for signature tuples and witnesses for small projections. The only significantly different subroutine is identified as the generalized version of nonempty, namely nonempty_gmm, which checks existence of appropriate witnesses for projections of size up to $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$2. The function fixvaluesgmm uses nonempty_gmm repeatedly.

The stated polynomial-time behavior is justified structurally rather than by a single closed-form bound. The source explains that the algorithm manipulates a polynomial number of layers, each layer has polynomial size, and the generation procedure uses only fixed-arity operations and bounded search over finite domains. For the GMM formalization it additionally states that, for each level, the number of calls is bounded by a polynomial such as $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$3, each nonempty_gmm call uses only $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$4 layers for the relevant arity $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$5, and the resulting sets are bounded by a polynomial in the number of variables (Gaysin, 1 Aug 2025).

This account also clarifies the algorithm's effective data model. Dalmau's Algorithm does not explicitly enumerate the full solution relation after each constraint insertion. It instead carries a compressed generating description whose correctness is measured by preservation of signature information and bounded-arity projections.

5. Formalization in $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$6 and extended Frege consequences

A central reinterpretation of Dalmau's Algorithm is proof-theoretic. The formalization is carried out in the two-sorted bounded-arithmetic theory $$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$7, which the source identifies with polynomial-time reasoning and links to the extended Frege system through the standard translation theorem (Gaysin, 1 Aug 2025).

The bounded-arithmetic objective is to prove soundness: if the algorithm returns “no,” then the instance is unsatisfiable. For the GMM case, the paper states:

$$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$8

The proof-complexity consequence is then given as:

$$\text{Theorem 6 ([13]). Let } A \text{ be a finite set and } y \text{ be a GMM operation on } A. \text{ Then } \mathrm{CSP}(\mathrm{Inv}(p)) \text{ is solvable in polynomial time.}$$9

that, on any unsatisfiable instance $(k+1)$0 of $(k+1)$1, outputs a polynomial-size EF proof of the propositional translation of $(k+1)$2.

The route from algorithmic soundness to propositional proof upper bounds is mediated by a general translation principle:

$(k+1)$3

In the CSP setting, one encodes an instance $(k+1)$4 and template $(k+1)$5 into a propositional formula $(k+1)$6 expressing existence of a homomorphism. When $(k+1)$7 is unsatisfiable, the negation is a tautology. The soundness theorem for Dalmau's Algorithm therefore yields short EF refutations of unsatisfiable GMM instances.

The proof idea follows the same pattern as in the Mal'tsev case: maintain a compact representation $(k+1)$8 after processing $(k+1)$9 constraints, prove inside $\{a,b\}$0 that next maps correct compact representations to correct compact representations, and conclude that an empty final representation precludes any homomorphism. The paper adds an important efficiency point from the Mal'tsev formalization: one does not need to prove that each $\{a,b\}$1 generates the full solution set; it is enough to prove that the signatures match and $\{a,b\}$2. It then states that the same overall pattern applies to Dalmau's Algorithm (Gaysin, 1 Aug 2025).

6. Related algebraic viewpoints and scope boundaries

The most immediate conceptual boundary is between Dalmau's Algorithm and the Bulatov–Dalmau algorithm for Mal'tsev CSPs. The proof-complexity paper presents the former as the GMM generalization of the latter: the methods are “very similar,” but the GMM version requires the more elaborate representation described above, because small projections up to size $\{a,b\}$3 must be preserved in addition to signature information (Gaysin, 1 Aug 2025).

A second boundary concerns recurrence-based and FGLM-style algebraic algorithms. The paper "Algorithms for zero-dimensional ideals using linear recurrent sequences" does not present Dalmau's Algorithm by name in the excerpted text, but it explicitly frames relevant connections. It describes an FGLM-style and annihilator-based environment built from monomial bases, multiplication matrices, linear dependence tests on generalized Hankel-type matrices, annihilator ideals of one or several sequences, and lexicographic recovery by border exploration. The same source states that these are the key shared objects for understanding Dalmau-type methods (Neiger et al., 2017).

That comparison is nonetheless limited. The zero-dimensional-ideal paper extends beyond a basic Dalmau-like change-of-order or single-component viewpoint by using one or several sequences rather than a single sequence, by handling non-Gorenstein situations in which one sequence does not suffice, and by splitting an ideal through factorization of a minimal polynomial,

$\{a,b\}$4

with component isolation via transposed products such as

$\{a,b\}$5

It therefore provides a broader annihilator-based toolkit rather than a direct presentation of Dalmau's Algorithm itself (Neiger et al., 2017).

In the narrower CSP setting, the bottom line is more specific. Dalmau's Algorithm is a polynomial-time algorithm for CSPs preserved by a GMM polymorphism; it is based on compact representations of solution sets; its correctness can be formalized in $\{a,b\}$6; and that soundness theorem yields polynomial-size extended Frege proofs for propositional unsatisfiability statements of the corresponding GMM instances (Gaysin, 1 Aug 2025).