Polynomial-Time Algorithms for MSC

• The paper establishes the tractability of Mal'tsev CSPs via a compact representation strategy that enables polynomial-time decision procedures.
• It details an incremental algorithm using the 'NEXT' subroutine and witness tuples to efficiently update constraint intersections.
• The approach is formalized in bounded arithmetic (V¹), ensuring both practical solution certification and short propositional proofs for unsolvable instances.

Polynomial-time algorithms for Mal'tsev CSP (MSC) represent a major research milestone in constraint satisfaction complexity and algorithm design. The class of Mal'tsev CSPs comprises all CSP templates invariant under a fixed Mal'tsev operation—an algebraic ternary operation $\mu$ on domain $A$ satisfying $\mu(x, y, y) = \mu(y, y, x) = x$ for all $x, y \in A$. The foundational work of Bulatov and Dalmau established the tractability of MSC through a compact-representation approach and provided an explicit polynomial-time algorithm for deciding and solving arbitrary MSC instances. The algorithmic framework has been further refined and formalized in the context of bounded arithmetic and proof complexity (Gaysin, 1 Aug 2025). This article presents the complete algorithmic, algebraic, and proof-theoretic aspects of polynomial-time MSC algorithms, culminating in their formal soundness within $V^1$ and feasibility as efficient propositional proof certificates.

1. Mal'tsev Operations, Invariant Relations, and Problem Definition

A Mal'tsev operation $\mu: A^3 \to A$ on a finite set $A$ is defined by the identities $\mu(x, y, y) = x$ and $\mu(y, y, x) = x$ for all $x, y \in A$. An $n$-ary relation $R \subseteq A^n$ is invariant under $\mu$ if for any three tuples $t^{(1)}, t^{(2)}, t^{(3)} \in R$, the tuple $u$ given by $u[i] = \mu(t^{(1)}[i], t^{(2)}[i], t^{(3)}[i])$ also lies in $R$. A CSP template is Mal'tsev if every base relation is invariant under some fixed $\mu$.

Given a CSP instance $P = (X, A, C)$, with $X = \{x_1, \ldots, x_n\}$ variables over domain $A$ and constraints $C = \{C_1, \ldots, C_m\}$ of the form $(S_l, R_l)$ where $S_l$ is a scope and $R_l \subseteq A^{|S_l|}$, the central computational task is to check whether $P$ admits a solution $\phi : X \to A$ mapping every constraint's scope into its relation. For Mal'tsev CSPs, all $R_l$ are invariant under a fixed $\mu$.

2. Compact Representations, Signatures, and Subalgebra Generation

The Bulatov–Dalmau algorithm exploits algebraic closure under $\mu$ by operating on compact representations. For any $R \subseteq A^n$, the signature $\operatorname{Sig}(R)$ is the set of all $(i, a, b)$ such that there exist $t, t' \in R$ with $t[1\,..\,i-1] = t'[1\,..\,i-1]$, $t[i]=a$, $t'[i]=b$. A subset $R' \subseteq R$ is a compact representation if $\operatorname{Sig}(R') = \operatorname{Sig}(R)$ and $|R'| \leq 2 |\operatorname{Sig}(R')|$. The central theorem is that for a Mal'tsev-invariant relation $R$ and any compact representation $R'$, the subalgebra generated by coordinate-wise application of $\mu$ to $R'$ is exactly $R$. This allows updating and maintaining polynomial-sized representations through the constraint intersection process.

3. Polynomial-time Algorithm: Incremental Constraint Propagation

The algorithm initializes with a compact representation for the unconstrained product $A^n$. For each constraint $C_\ell$ in sequence, the method produces a compact representation for the intersection $$R_\ell = R_{\ell-1} \cap \{\text{assignments satisfying }C_\ell\}$$ using the "NEXT" subroutine. For binary constraints $(i, j)$ and relation $S \subseteq A^2$, "NEXT" computes, for each signature entry, witness tuples and applies $\mu$ to ensure closure under the Mal'tsev operation.

Pseudo-code for iteration:

Algorithm MSC-Solver(P) Input: CSP instance P=(X, A, C) Output: Solution φ or "no solution" 1. R^(0) ← compact representation of A^n 2. For ℓ = 1 to m: R^(ℓ) ← NEXT(R^(ℓ-1), (i₁,...,i_s), S_ℓ) 3. If R^(m) ≠ ∅: return solution from R^(m) Else: return "no solution"

Each "NEXT" call requires $O(n|A|^2)$ signature entries, with witness-finding and $\mu$ evaluations taking polynomial time; overall worst-case time is $O(m n^8 + m(n + |S^*|)^4 |S^*| n^2)$ (Gaysin, 1 Aug 2025).

4. Correctness, Soundness, and Completeness

The method ensures that at each stage, the compact representation $R^{(\ell)}$ generates precisely the set of partial solutions respecting the first $\ell$ constraints. If at the final step $R^{(m)}$ is nonempty, a valid assignment is extractable. If $R^{(m)}$ is empty, the instance has no solution. Soundness, witnessed by the preservation of signatures and generation theorem, is critical for both practical correctness and formal proof complexity consequences.

5. Formalization in Bounded Arithmetic and Proof Complexity

The algorithm for Mal'tsev CSPs is fully formalizable in the theory $V^1$, which corresponds to polynomial-time computations and captures the extended Frege ($\mathsf{EF}$) propositional proof system. Each stage (encoding, witness search, application of $\mu$, verification of constraints) can be expressed in $\Sigma^B_1$ formulas with induction bounded by polynomial size. The soundness of the algorithm (i.e., that unsatisfiability yields a certificate of nonexistence of solutions) is provable inside $V^1$ by bounded induction. Consequently, the propositional formula encoding "no homomorphism exists" for an unsatisfiable Mal'tsev CSP admits a polynomial-size $\mathsf{EF}$ proof (Gaysin, 1 Aug 2025).

This alignment between algorithmic tractability and proof complexity establishes that the Boolean tautologies expressing unsatisfiability of Mal'tsev CSPs have short propositional proofs, thus bridging algebraic tractability and efficient certification.

6. Extensions: Generalized Majority-Minority CSPs

The framework encompasses Dalmau's algorithm for CSPs invariant under generalized majority-minority (GMM) operations, an expansion strictly containing Mal'tsev and near-unanimity CSPs. For GMM-CSPs, the compact representation and signature propagation are analogous, but must also track projections onto all $k$-sized coordinate subsets and handle minority/majority pairs. The algorithm and its soundness remain formalizable within $V^1$, yielding similar proof complexity guarantees for these broader tractable CSP subclasses.

7. Connections and Significance

Polynomial-time algorithms for Mal'tsev CSPs establish the precise computational boundary between algebraically tractable and NP-complete constraint satisfaction templates, as dictated by the CSP dichotomy theorem. The explicit compact representation approach not only delivers efficient solvers but also provides a uniform mechanism for proof certification of unsatisfiability. The $V^1$ formalization ensures that every non-solvable instance admits efficient certificate extraction and short propositional refutations, serving both algorithmic applications and foundational results in proof complexity. The methodology applies equally to generalized majority-minority CSPs, further illuminating the structural landscape of tractable constraint satisfaction algorithms.