On Boolean PCSPs with Polynomial Threshold Polymorphisms

Katzper Michno

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arXiv · 2509.26248

On Boolean PCSPs with Polynomial Threshold Polymorphisms

Katzper Michno

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Abstract

In pursuit of a deeper understanding of Boolean Promise Constraint Satisfaction Problems (PCSPs), we identify a class of problems with restricted structural complexity, which could serve as a promising candidate for complete characterization. Specifically, we investigate the class of PCSPs whose polymorphisms are Polynomial Threshold Functions (PTFs) of bounded degree. We obtain two complexity characterization results: (1) with a hardness condition introduced in [ACMTCT'21], we establish a complete complexity dichotomy in the case where coefficients of PTF representations are non-negative; (2) dropping the non-negativity assumption, we show a hardness result for PTFs admitting coordinates with significant influence, conditioned on the Rich 2-to-1 Conjecture proposed in [ITCS'21]. In order to prove the latter, we show that a random 2-to-1 minor map retains significant coordinate influence over the $p$-biased hypercube with constant probability.

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Katzper Michno. 2025-09-30. On Boolean PCSPs with Polynomial Threshold Polymorphisms. https://arxiv.org/abs/2509.26248

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