{"title":"近乎精确的垃圾箱覆盖问题","authors":"Asaf Levin","doi":"10.1007/s00453-024-01224-5","DOIUrl":null,"url":null,"abstract":"<div><p>We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant <span>\\(\\varDelta \\)</span>, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and <span>\\(1+\\varDelta \\)</span>. Our goal is to maximize the number of near exact covered bins. If <span>\\(\\varDelta =0\\)</span> or <span>\\(\\varDelta >0\\)</span> is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that <span>\\(P\\ne NP\\)</span>). However, for the case where <span>\\(\\varDelta >0\\)</span> is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 6","pages":"2041 - 2066"},"PeriodicalIF":0.9000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01224-5.pdf","citationCount":"0","resultStr":"{\"title\":\"The Near Exact Bin Covering Problem\",\"authors\":\"Asaf Levin\",\"doi\":\"10.1007/s00453-024-01224-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant <span>\\\\(\\\\varDelta \\\\)</span>, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and <span>\\\\(1+\\\\varDelta \\\\)</span>. Our goal is to maximize the number of near exact covered bins. If <span>\\\\(\\\\varDelta =0\\\\)</span> or <span>\\\\(\\\\varDelta >0\\\\)</span> is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that <span>\\\\(P\\\\ne NP\\\\)</span>). However, for the case where <span>\\\\(\\\\varDelta >0\\\\)</span> is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.</p></div>\",\"PeriodicalId\":50824,\"journal\":{\"name\":\"Algorithmica\",\"volume\":\"86 6\",\"pages\":\"2041 - 2066\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00453-024-01224-5.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algorithmica\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00453-024-01224-5\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algorithmica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00453-024-01224-5","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant \(\varDelta \), and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and \(1+\varDelta \). Our goal is to maximize the number of near exact covered bins. If \(\varDelta =0\) or \(\varDelta >0\) is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that \(P\ne NP\)). However, for the case where \(\varDelta >0\) is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.
期刊介绍：
Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential.
Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming.
In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.
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