{"title":"PCSP 的注入硬度条件","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":"46 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"46 1\",\"pages\":\"\"},\"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}","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}
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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