{"title":"通用因子图的最优不可逼近性","authors":"Per Austrin, Jonah Brown-Cohen, Johan Håstad","doi":"10.1145/3631119","DOIUrl":null,"url":null,"abstract":"<p>The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance. </p><p>Examples of results obtained for this restricted setting are: <p><table border=\"0\" list-type=\"ordered\" width=\"95%\"><tr><td valign=\"top\"><p>(1)</p></td><td colspan=\"5\" valign=\"top\"><p>Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001).</p></td></tr><tr><td valign=\"top\"><p>(2)</p></td><td colspan=\"5\" valign=\"top\"><p>Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016).</p></td></tr><tr><td valign=\"top\"><p>(3)</p></td><td colspan=\"5\" valign=\"top\"><p>Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017).</p></td></tr></table></p>\nThe main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimal Inapproximability with Universal Factor Graphs\",\"authors\":\"Per Austrin, Jonah Brown-Cohen, Johan Håstad\",\"doi\":\"10.1145/3631119\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance. </p><p>Examples of results obtained for this restricted setting are: <p><table border=\\\"0\\\" list-type=\\\"ordered\\\" width=\\\"95%\\\"><tr><td valign=\\\"top\\\"><p>(1)</p></td><td colspan=\\\"5\\\" valign=\\\"top\\\"><p>Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001).</p></td></tr><tr><td valign=\\\"top\\\"><p>(2)</p></td><td colspan=\\\"5\\\" valign=\\\"top\\\"><p>Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016).</p></td></tr><tr><td valign=\\\"top\\\"><p>(3)</p></td><td colspan=\\\"5\\\" valign=\\\"top\\\"><p>Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017).</p></td></tr></table></p>\\nThe main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.</p>\",\"PeriodicalId\":50922,\"journal\":{\"name\":\"ACM Transactions on Algorithms\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-12-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Algorithms\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1145/3631119\",\"RegionNum\":3,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Algorithms","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3631119","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Optimal Inapproximability with Universal Factor Graphs
The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.
Examples of results obtained for this restricted setting are:
(1)
Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001).
(2)
Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016).
(3)
Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017).
The main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.
期刊介绍：
ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include
combinatorial searches and objects;
counting;
discrete optimization and approximation;
randomization and quantum computation;
parallel and distributed computation;
algorithms for
graphs,
geometry,
arithmetic,
number theory,
strings;
on-line analysis;
cryptography;
coding;
data compression;
learning algorithms;
methods of algorithmic analysis;
discrete algorithms for application areas such as
biology,
economics,
game theory,
communication,
computer systems and architecture,
hardware design,
scientific computing
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