{"title":"NP-Hard SAT和CSP问题的(粗)细晶结构","authors":"Victor Lagerkvist, Magnus Wahlström","doi":"10.1145/3492336","DOIUrl":null,"url":null,"abstract":"We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis(SETH), showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c0 > 1 such that the problem cannot be solved in time O(cn) for any c <c0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O(cn) for some c< 2. Such lower bounds have proven extremely elusive, and except for cases where c0=2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations. Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"18 2 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems\",\"authors\":\"Victor Lagerkvist, Magnus Wahlström\",\"doi\":\"10.1145/3492336\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis(SETH), showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c0 > 1 such that the problem cannot be solved in time O(cn) for any c <c0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O(cn) for some c< 2. Such lower bounds have proven extremely elusive, and except for cases where c0=2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations. Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.\",\"PeriodicalId\":198744,\"journal\":{\"name\":\"ACM Transactions on Computation Theory (TOCT)\",\"volume\":\"18 2 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Computation Theory (TOCT)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3492336\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Computation Theory (TOCT)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3492336","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems
We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis(SETH), showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c0 > 1 such that the problem cannot be solved in time O(cn) for any c
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