{"title":"Injective hardness condition for PCSPs","authors":"Demian Banakh, Marcin Kozik","doi":"arxiv-2405.10774","DOIUrl":null,"url":null,"abstract":"We present a template for the Promise Constraint Satisfaction Problem (PCSP)\nwhich is NP-hard but does not satisfy the current state-of-the-art hardness\ncondition [ACMTCT'21]. We introduce a new \"injective\" condition based on the\nsmooth version of the layered PCP Theorem and use this new condition to confirm\nthat the problem is indeed NP-hard. In the second part of the article, we\nestablish a dichotomy for Boolean PCSPs defined by templates with polymorphisms\nin the set of linear threshold functions. The reasoning relies on the new\ninjective condition.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Computational Complexity","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.10774","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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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