Injective hardness condition for PCSPs
Demian Banakh, Marcin Kozik
TL;DR
This paper introduces an injective hardness condition for PCSPs, demonstrating NP-hardness for certain problems and establishing a dichotomy for Boolean PCSPs with specific polymorphisms, advancing the understanding of PCSP complexity.
Contribution
It proposes a new injective condition based on layered PCPs, expanding the tools for proving PCSP NP-hardness and classifying Boolean PCSPs.
Findings
Confirmed NP-hardness of certain PCSPs using the new injective condition.
Established a dichotomy for Boolean PCSPs with linear threshold polymorphisms.
Demonstrated the injective condition's effectiveness beyond previous hardness criteria.
Abstract
We present a template for the Promise Constraint Satisfaction Problem (PCSP) which is NP-hard but does not satisfy the current state-of-the-art hardness condition [ACMTCT'21]. We introduce a new "injective" condition based on the smooth version of the layered PCP Theorem and use this new condition to confirm that the problem is indeed NP-hard. In the second part of the article, we establish a dichotomy for Boolean PCSPs defined by templates with polymorphisms in the set of linear threshold functions. The reasoning relies on the new injective condition.
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