{"title":"具有部分处理时间信息的两个层次均匀机器上的半在线早期工作最大化问题","authors":"Man Xiao, Xiaoqiao Liu, Weidong Li","doi":"10.1007/s10878-023-01086-7","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we consider four semi-online early work maximization problems on two hierarchical uniform machines <span>\\(M_1\\)</span> and <span>\\(M_2\\)</span>, where machine <span>\\(M_1\\)</span> with speed <span>\\(s>0\\)</span> is available for all jobs and machine <span>\\(M_2\\)</span> with speed 1 is only available for high-hierarchy jobs. When the total size of all jobs is known, we design an optimal online algorithm with a competitive ratio of <span>\\(\\min \\{1+s,\\frac{2+2s}{1+2s}\\}\\)</span>. When the total size of low-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of <span>\\(\\min {\\{1+s, \\frac{\\sqrt{9\\,s^2+10\\,s+1}-s-1}{2\\,s}}\\}\\)</span>. When the total size of high-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of <span>\\(\\min \\{\\sqrt{s+1}, \\sqrt{s^2+2\\,s+2}-s\\}\\)</span>. When both the total sizes of low-hierarchy and high-hierarchy jobs are known, we give a lower bound <span>\\(\\frac{2s+2}{s+2}\\)</span> for the case <span>\\(s\\le \\frac{2}{3}\\)</span>, and an optimal online algorithm with a competitive ratio of <span>\\(\\frac{3s+3}{3s+2}\\)</span> for the case <span>\\(s>\\frac{2}{3}\\)</span>.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Semi-online early work maximization problems on two hierarchical uniform machines with partial information of processing time\",\"authors\":\"Man Xiao, Xiaoqiao Liu, Weidong Li\",\"doi\":\"10.1007/s10878-023-01086-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we consider four semi-online early work maximization problems on two hierarchical uniform machines <span>\\\\(M_1\\\\)</span> and <span>\\\\(M_2\\\\)</span>, where machine <span>\\\\(M_1\\\\)</span> with speed <span>\\\\(s>0\\\\)</span> is available for all jobs and machine <span>\\\\(M_2\\\\)</span> with speed 1 is only available for high-hierarchy jobs. When the total size of all jobs is known, we design an optimal online algorithm with a competitive ratio of <span>\\\\(\\\\min \\\\{1+s,\\\\frac{2+2s}{1+2s}\\\\}\\\\)</span>. When the total size of low-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of <span>\\\\(\\\\min {\\\\{1+s, \\\\frac{\\\\sqrt{9\\\\,s^2+10\\\\,s+1}-s-1}{2\\\\,s}}\\\\}\\\\)</span>. When the total size of high-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of <span>\\\\(\\\\min \\\\{\\\\sqrt{s+1}, \\\\sqrt{s^2+2\\\\,s+2}-s\\\\}\\\\)</span>. When both the total sizes of low-hierarchy and high-hierarchy jobs are known, we give a lower bound <span>\\\\(\\\\frac{2s+2}{s+2}\\\\)</span> for the case <span>\\\\(s\\\\le \\\\frac{2}{3}\\\\)</span>, and an optimal online algorithm with a competitive ratio of <span>\\\\(\\\\frac{3s+3}{3s+2}\\\\)</span> for the case <span>\\\\(s>\\\\frac{2}{3}\\\\)</span>.</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-10-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-023-01086-7\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01086-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Semi-online early work maximization problems on two hierarchical uniform machines with partial information of processing time
In this paper, we consider four semi-online early work maximization problems on two hierarchical uniform machines \(M_1\) and \(M_2\), where machine \(M_1\) with speed \(s>0\) is available for all jobs and machine \(M_2\) with speed 1 is only available for high-hierarchy jobs. When the total size of all jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min \{1+s,\frac{2+2s}{1+2s}\}\). When the total size of low-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min {\{1+s, \frac{\sqrt{9\,s^2+10\,s+1}-s-1}{2\,s}}\}\). When the total size of high-hierarchy jobs is known, we design an optimal online algorithm with a competitive ratio of \(\min \{\sqrt{s+1}, \sqrt{s^2+2\,s+2}-s\}\). When both the total sizes of low-hierarchy and high-hierarchy jobs are known, we give a lower bound \(\frac{2s+2}{s+2}\) for the case \(s\le \frac{2}{3}\), and an optimal online algorithm with a competitive ratio of \(\frac{3s+3}{3s+2}\) for the case \(s>\frac{2}{3}\).
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.