The complexity of finding coset-generating polymorphisms and the promise metaproblem

We show that the metaproblem for coset-generating polymorphisms is NP-complete, answering a question of Chen and Larose: given a finite structure, the computational question is whether this structure has a polymorphism of the form $(x,y,z) \mapsto x y^{-1} z$ with respect to some group; such operations are also called coset-generating, or heaps. Furthermore, we introduce a promise version of the metaproblem, parametrised by two polymorphism conditions $Σ_1$ and $Σ_2$ and defined analogously to the promise constraint satisfaction problem. We give sufficient conditions under which the promise metaproblem for $(Σ_1,Σ_2)$ is in P and under which it is NP-hard. In particular, the promise metaproblem is in P if $Σ_1$ states the existence of a Maltsev polymorphism and $Σ_2$ states the existence of an abelian heap polymorphism – despite the fact that neither the metaproblem for $Σ_1$ nor the metaproblem for $Σ_2$ is known to be in P. We also show that the creation-metaproblem for Maltsev polymorphisms, under the promise that a heap polymorphism exists, is in P if and only if there is a uniform polynomial-time algorithm for CSPs with a heap polymorphism.

💡 Research Summary

The paper investigates the computational complexity of determining whether a finite relational structure admits a coset‑generating polymorphism (also known as a heap), an operation of the form f(x,y,z)=x y⁻¹ z for some group. This question, posed by Chen and Larose, is formalized as a meta‑problem: given a structure B (with relations listed explicitly), decide whether its polymorphism clone Pol(B) contains such an operation. The authors prove that this meta‑problem is NP‑complete. Their hardness reduction encodes an arbitrary 3‑SAT instance into a structure built from direct products of groups of order 4p (p≥5 prime). The construction ensures that a coset‑generating polymorphism exists exactly when the original formula is satisfiable, establishing NP‑hardness; membership in NP is straightforward, yielding NP‑completeness. Consequently, any uniform polynomial‑time algorithm for CSPs with coset‑generating polymorphisms cannot rely on first computing the polymorphism and then applying the semi‑uniform algorithm of Bulatov‑Dalmau.

Beyond this core result, the authors introduce a promise version of the meta‑problem, denoted PMeta(Σ₁,Σ₂). Here Σ₁ and Σ₂ are polymorphism conditions (sets of identities) with Σ₁ ⇒ Σ₂. The promise problem asks, for a given structure B, to decide whether Pol(B) satisfies Σ₁ or fails even Σ₂. This mirrors the framework of Promise CSPs but is applied to polymorphism existence rather than homomorphism existence.

The paper’s most striking tractability result is that when Σ₁ requires a Maltsev polymorphism (an operation m satisfying m(x,x,y)=m(y,x,x)=y) and Σ₂ requires an abelian heap (a coset‑generating operation derived from an abelian group), the promise meta‑problem lies in P. The key insight is that the existence of an abelian heap enables the use of an affine integer programming (AIP) based uniform algorithm for CSPs with such polymorphisms. Using this algorithm, one can either construct a Maltsev polymorphism in polynomial time or certify that no abelian heap exists. Thus, despite the fact that the individual meta‑problems for Maltsev polymorphisms and for abelian heaps are not known to be in P, their promise combination is efficiently decidable.

The authors further generalize this phenomenon. They prove that for any idempotent Σ₁ that admits a uniform polynomial‑time CSP algorithm and any linear strong Maltsev condition Σ₂ (a height‑one strong Maltsev condition), PMeta(Σ₁,Σ₂) is in P provided a semi‑uniform algorithm exists for CSPs satisfying Σ₂. Moreover, under the same assumptions, the uniform CSP for Σ₁ is in P if and only if the “promise creation‑meta‑problem” (the task of actually constructing a polymorphism under the promise that a Σ₂‑polymorphism exists) is in P. This yields a precise equivalence: a uniform polynomial‑time algorithm for CSPs with coset‑generating polymorphisms exists exactly when the promise creation‑meta‑problem for coset‑generating versus Maltsev polymorphisms is solvable in polynomial time.

On the hardness side, the paper extends a result of Chen and Larose: if Σ₁ and Σ₂ are non‑trivial, consistent, height‑one strong Maltsev conditions with Σ₁ ⇒ Σ₂, then PMeta(Σ₁,Σ₂) is NP‑complete. This shows that the promise framework does not automatically soften the difficulty; the complexity hinges on the algebraic strength of the conditions.

The paper also discusses the relationship between the meta‑problem and the “creation‑meta‑problem” (finding an explicit polymorphism). For coset‑generating polymorphisms, the meta‑problem is NP‑complete, but the creation‑meta‑problem is at least as hard. However, when an abelian heap is promised, the creation‑meta‑problem for Maltsev polymorphisms becomes tractable, linking the two problems tightly.

In the concluding section, the authors list open questions: (i) whether a uniform polynomial‑time algorithm for CSPs with arbitrary coset‑generating polymorphisms exists, (ii) the exact complexity of the creation‑meta‑problem for Maltsev polymorphisms without any promise, and (iii) a full classification of promise meta‑problems for broader families of polymorphism conditions.

Overall, the paper makes three major contributions: (1) establishing NP‑completeness of the coset‑generating polymorphism meta‑problem, (2) introducing and analyzing the promise meta‑problem, providing both tractability and hardness criteria, and (3) revealing deep connections between uniform CSP algorithms, meta‑problems, and their promise variants, thereby advancing our understanding of the algebraic approach to CSP complexity.