{"title":"二维矢量多重背包的改进近似值","authors":"","doi":"10.1016/j.comgeo.2024.102124","DOIUrl":null,"url":null,"abstract":"<div><p>We study the <span>uniform</span> 2<span>-dimensional vector multiple knapsack</span> (2VMK) problem, a natural variant of <span>multiple knapsack</span> arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional <em>weight</em> vector and a positive <em>profit</em>, along with <em>m</em> 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.</p><p>Our main result is a <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ln</mi><mo></mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm for 2VMK, for every fixed <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, thus improving the best known ratio of <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span> which follows as a special case from a result of Fleischer et al. (2011) <span><span>[6]</span></span>.</p><p>Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) <span><span>[15]</span></span>, originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to <span><math><mo>≈</mo><mi>m</mi><mo>⋅</mo><mi>ln</mi><mo></mo><mn>2</mn><mo>≈</mo><mn>0.693</mn><mo>⋅</mo><mi>m</mi></math></span> of the bins, followed by a reduction to the (1-dimensional) <span>Multiple Knapsack</span> problem for assigning items to the remaining bins.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000464/pdfft?md5=aabf82f5f8cf463934bfaf0d08024ae5&pid=1-s2.0-S0925772124000464-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Improved approximation for two-dimensional vector multiple knapsack\",\"authors\":\"\",\"doi\":\"10.1016/j.comgeo.2024.102124\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study the <span>uniform</span> 2<span>-dimensional vector multiple knapsack</span> (2VMK) problem, a natural variant of <span>multiple knapsack</span> arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional <em>weight</em> vector and a positive <em>profit</em>, along with <em>m</em> 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.</p><p>Our main result is a <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ln</mi><mo></mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm for 2VMK, for every fixed <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, thus improving the best known ratio of <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span> which follows as a special case from a result of Fleischer et al. (2011) <span><span>[6]</span></span>.</p><p>Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) <span><span>[15]</span></span>, originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to <span><math><mo>≈</mo><mi>m</mi><mo>⋅</mo><mi>ln</mi><mo></mo><mn>2</mn><mo>≈</mo><mn>0.693</mn><mo>⋅</mo><mi>m</mi></math></span> of the bins, followed by a reduction to the (1-dimensional) <span>Multiple Knapsack</span> problem for assigning items to the remaining bins.</p></div>\",\"PeriodicalId\":51001,\"journal\":{\"name\":\"Computational Geometry-Theory and Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0925772124000464/pdfft?md5=aabf82f5f8cf463934bfaf0d08024ae5&pid=1-s2.0-S0925772124000464-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Geometry-Theory and Applications\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0925772124000464\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Geometry-Theory and Applications","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0925772124000464","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Improved approximation for two-dimensional vector multiple knapsack
We study the uniform 2-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional weight vector and a positive profit, along with m 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.
Our main result is a -approximation algorithm for 2VMK, for every fixed , thus improving the best known ratio of which follows as a special case from a result of Fleischer et al. (2011) [6].
Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) [15], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.
期刊介绍:
Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems.
Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.