Boolean Promise CSPs: Complexity & Algorithms

• Boolean Promise CSPs are generalized constraint satisfaction problems that encode approximation challenges by distinguishing strict from relaxed satisfiability.
• Key dichotomy results show that the presence of robust polymorphism families enables polynomial-time tractability, while their absence implies NP-hardness.
• Advanced algorithmic techniques, including LP, SDP, and singleton methods, leverage algebraic symmetries to solve or prove intractability in these problems.

Boolean Promise Constraint Satisfaction Problems (PCSPs) constitute a rich and highly structured generalization of classical Boolean CSPs, central to recent advances in computational complexity, approximation algorithms, and universal algebra. Boolean PCSPs formalize a decision gap between strict and relaxed forms of constraints, encapsulating many natural approximation problems (for example, approximate graph and hypergraph coloring, variants of SAT, and discrepancy minimization) within a common algebraic and algorithmic framework. Research over the past decade has produced deep dichotomy theorems under various symmetry assumptions, introduced sophisticated relaxation and rounding methodologies, and revealed intricate algebraic and categorical underpinnings driving both tractability and intractability.

1. Formal Definition and Structural Foundations

A Boolean Promise Constraint Satisfaction Problem is defined by a pair of Boolean relational templates $(A, B)$ over a common signature, together with the requirement $A \rightarrow B$ (that is, there exists a homomorphism from $A$ to $B$). A PCSP instance $(X)$ is a finite relational structure of the same signature, with the promise that either there is a homomorphism $X \to A$ (“strict” satisfiability) or there is no homomorphism from $X \to B$ (“relaxed” unsatisfiability). The algorithmic challenge is: given such an $X$, decide whether $X \to B$ (computational search variants are also considered).

This framework generalizes classical Boolean CSPs: if $A = B$ the PCSP collapses to standard CSP. The addition of the promise separates PCSPs from mere local consistency or relaxation-based approaches and means that Boolean PCSPs encode a diverse family of approximation, partial coloring, and satisfaction problems. For a template with $A$ and $B$ both “symmetric” (i.e., relations invariant under coordinate permutation), fundamental examples include $$(1\text{-in-}3\text{-SAT}, \text{NAE-}3\text{-SAT})$$ and $(k\text{-coloring}, l\text{-coloring})$ for graphs and hypergraphs (Brakensiek et al., 2017, Barto et al., 2020).

A key algebraic object is the minion of polymorphisms, $\operatorname{Pol}(A,B)$: the set of all functions $f : A^n \to B$ (for all $n \geq 1$) that, when applied coordinatewise to tuples from a constraint $P \subseteq A^k$, yield an output in the relaxed constraint $Q \subseteq B^k$ for every $(P,Q)$ constraint pair. Unlike ordinary CSPs, these weak polymorphisms do not generally form clones; composition may not be closed (Brakensiek et al., 2017, Hadek et al., 13 Mar 2025).

2. Algebraic Approach and Dichotomy Theorems

The tractability of Boolean PCSPs is determined by the algebraic structure of their polymorphisms. A central result is a dichotomy for symmetric Boolean PCSPs (with or without negation):

• Tractability: If $\operatorname{Pol}(A,B)$ contains polymorphisms from certain robust families—such as Parity, Majority, Alternating Threshold ($\operatorname{AT}$), $\operatorname{XOR}$, Max, Min, or rational-threshold functions (and their complements)—for all sufficiently large arities, then the PCSP is solvable in polynomial time, typically via linear programming, Gaussian elimination, or basic affine integer programming (Brakensiek et al., 2017, Ficak et al., 2019, Nakajima et al., 2022, Banakh et al., 17 May 2024).
• Hardness: If none of these polymorphisms are present, every polymorphism is “$C$‑fixing” (i.e., its output depends only on $C$ coordinates for some finite $C$), yielding inapproximability: one constructs a reduction from Label Cover or Gap Label Cover, establishing NP-hardness.

This dichotomy is sharpened by analyzing polymorphism minions for “junta-like” structure and by employing enhancements such as the injective layered choice condition, which provides a more flexible and encompassing criterion for intractability by relaxing the behavior required along chains of minor maps while still carrying weight in hardness reductions (Banakh et al., 17 May 2024).

3. Algorithmic Frameworks and Algebraic Symmetries

The algorithmic landscape for Boolean PCSPs is tightly linked to the symmetry properties of polymorphisms:

• Symmetric and Block-Symmetric Polymorphisms: Infinite families of symmetric polymorphisms guarantee tractability via algorithms combining the Basic Linear Programming (BLP) relaxation and the Affine Integer Programming (AIP) relaxation (Brakensiek et al., 2019). Block-symmetric polymorphisms—where symmetry is required within coordinate blocks—permit the same algorithm to succeed; block size “width” can be increased arbitrarily.
• Homomorphic Sandwiches: Many tractable Boolean PCSPs reduce via a sandwich $A \to C \to B$ to a tractable CSP($C$), often over an infinite domain (e.g., CSPs as systems of linear equations over $\mathbb{Z}$) (Brakensiek et al., 2018, Barto, 2019, Deng et al., 2020). When finite sandwich reductions exist, the intermediate $C$ can be arbitrarily large—even for very small $A$ and $B$—and may only exist beyond Boolean size (Kazda et al., 2021).
• CLAP and Singleton Algorithms: The recent CLAP algorithm (combining constraint-local BLP with AIP) strictly generalizes previous methods, solving PCSPs not handled by BLP+AIP or finite sandwich approaches. It exploits “$H$-symmetric” polymorphism families, which require symmetry only for tieless input profiles determined by a tie matrix $H$ (Ciardo et al., 2021). Singleton algorithms further strengthen propagation and are characterized algebraically by the existence of palette block symmetric polymorphisms, which are constructed using the Hales–Jewett theorem (Zhuk, 22 Sep 2025).

Algorithms are “polymorphism-driven”: tractability is achieved only in the presence of polymorphisms with high-dimensional symmetry or noise stability, while the absence of such polymorphisms leads to intractability by minor conditions or robust reductions. Notably, for symmetric and block-symmetric settings, all tractable Boolean CSPs (as classified by Schaefer’s dichotomy) are solved by these frameworks (Brakensiek et al., 2019, Nakajima et al., 2022).

4. Robustness, Relaxations, and Minion Characterization

Robust satisfaction, where nearly-satisfiable instances must yield nearly-satisfying assignments, prompts the paper of rounding from convex relaxations (especially semi-definite programming, SDP). SDP-based robust algorithms are feasible for Boolean PCSPs whose minion contains majority or alternating-threshold polymorphisms of all odd arities (Brakensiek et al., 2022). When these are absent, SDP integrality gaps and Unique Games-type hardness can be established using colorings of the sphere and arguments from sphere Ramsey theory.

A minion-theoretic characterization is available for the power of LP and SDP relaxations: there exists a minion homomorphism from the abstract LP or SDP minion (collections of vector or distributional solutions with minor operations defined) to $\operatorname{Pol}(A,B)$ if and only if the relaxation algorithm solves PCSP$(A, B)$ (Brakensiek et al., 2022, Ciardo et al., 2021, Brakensiek et al., 2019). This algebraic viewpoint is essential for comparing the power of relaxations (e.g., BLP+AIP is no more powerful than AIP for balanced templates (Nakajima et al., 2022)) and for understanding the limitations and strengths of universal algorithms.

The analytic structure of Boolean polymorphisms is further illuminated via influences, the Shapley value, and the paper of polynomial threshold functions (PTFs) and their minors. For PCSPs whose polymorphisms are (positive) PTFs of bounded degree, a complete complexity dichotomy is established through algebraic and isoperimetric arguments; in the more general PTF setting, NP-hardness is shown under the Rich 2-to-1 Conjecture when every polymorphism has a significantly influential coordinate (Michno, 30 Sep 2025).

5. Generalizations, Reductions, and Categorical Reformulations

PCSPs extend well beyond symmetric or Boolean instances. Classification results exist for several symmetric and functional PCSPs on non-Boolean domains, including nearly-complete treatments for hypergraph coloring where the admissibility relation is symmetric or functional (Barto et al., 2020, Nakajima et al., 2022). Complexity boundaries for “non-symmetric” problems have been studied using robust algebraic and analytic tests, and subtle changes to symmetric templates (for example, adding or removing a single tuple) can cause a transition between polynomial-time solvability and NP-hardness (Brandts et al., 2021).

All known tractable PCSPs reduce, sometimes via infinite or large finite intermediates, to tractable CSPs. For some classes, the size of the necessary “sandwiched” CSP can grow arbitrarily compared to the original template sizes, demonstrating that the tractable fragment of PCSPs is not always finitely witnessed (Kazda et al., 2021, Asimi et al., 2020, Deng et al., 2020).

Category-theoretic approaches provide a new unifying language: polymorphisms correspond to functors (e.g., as right Kan extensions), gadget reductions are interpreted as Yoneda extensions or nerves, and minion morphisms encapsulate adjunctions across these universes (Hadek et al., 13 Mar 2025). This reframing facilitates more abstract and general proofs and enables the import of techniques from algebraic topology into universal-algebraic complexity theory.

6. Frontier Problems and Open Directions

Despite comprehensive dichotomy theorems for broad classes (e.g., symmetric Boolean PCSPs), the general (non-symmetric) Boolean and non-Boolean cases remain open. Current hardness criteria (including the injective layered choice condition) are known to be essentially optimal for minions of linear threshold polymorphisms but completeness for, say, all bounded-degree polynomial threshold polymorphisms is not established outside the positive (nonnegative coefficient) case (Banakh et al., 17 May 2024, Michno, 30 Sep 2025). Classification for “ordered” Boolean PCSPs (where monotone polymorphisms are present) is known modulo the Rich 2-to-1 Conjecture via influence-based criteria (Brakensiek et al., 2021).

Robust dichotomies for satisfaction under relaxations (especially semidefinite programming) are known in special cases where minion homomorphisms exist from the SDP minion to $\operatorname{Pol}(A,B)$ (Brakensiek et al., 2022), but a characterization for arbitrary Boolean PCSPs is not yet resolved.

The harmonization of algebraic, analytic, and categorical insights continues to push the field forward. Future research aims at full dichotomy theorems for all Boolean PCSPs, further minion-theoretic characterizations of algorithmic power, and leveraging categorical methods for tractability and intractability proofs, with the ultimate goal of matching the depth and completeness of Schaefer’s dichotomy theorem in the broader PCSP context.

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Summary of Key Dichotomy Criteria for Symmetric Boolean PCSPs

Polymorphism Family Present	Algorithmic Consequence
Parity, Majority, Alternating-Threshold, rational-THR	Polynomial-time tractable
None of the above (e.g., all polymorphisms $C$-fixing)	NP-hard

Tractability requires the presence of robust, globally averaging polymorphisms for all odd arities; absence implies a structural collapse (fixing by small sets), which supports reduction-based NP-hardness (Brakensiek et al., 2017, Ficak et al., 2019, Banakh et al., 17 May 2024).

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Notable References

• "Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy" (Brakensiek et al., 2017)
• "Dichotomy for symmetric Boolean PCSPs" (Ficak et al., 2019)
• "The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs" (Brakensiek et al., 2019)
• "Injective hardness condition for PCSPs" (Banakh et al., 17 May 2024)
• "On Boolean PCSPs with Polynomial Threshold Polymorphisms" (Michno, 30 Sep 2025)
• "Singleton algorithms for the Constraint Satisfaction Problem" (Zhuk, 22 Sep 2025)
• "SDPs and Robust Satisfiability of Promise CSP" (Brakensiek et al., 2022)
• "On the complexity of symmetric vs. functional PCSPs" (Nakajima et al., 2022)
• "A categorical perspective on constraint satisfaction: The wonderland of adjunctions" (Hadek et al., 13 Mar 2025)