Injective hardness condition for PCSPs

TL;DR

This work targets the hardness frontier of Promise Constraint Satisfaction Problems (PCSPs). It introduces an injective layered hardness condition derived from a smooth, layered Label Cover framework and uses it to prove NP-hardness for a new Boolean PCSP template not captured by prior criteria. Additionally, it establishes a dichotomy for Boolean PCSP templates whose polymorphisms are linear-threshold functions, clarifying tractability versus hardness within this family. The results advance a unified approach to hardness in PCSPs and illuminate the boundary between tractable and intractable Boolean PCSPs, with implications for broader classes of PCSP templates and their algebraic structure.

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.

Theorems & Definitions (72)

• Theorem 1.1: The CSP dichotomy theorem
• Theorem 1.2
• Definition : decision CSP
• Definition : search CSP
• Definition : decision PCSP
• Definition : search PCSP
• Theorem 2.1: See Theorem 6.2 in brakensiek2017promise
• Definition : Minor, minor map
• Theorem 3.1
• Theorem 3.2