{"title":"固定维度下最小特征值最大化的快速算法","authors":"Adam Brown , Aditi Laddha , Mohit Singh","doi":"10.1016/j.orl.2024.107186","DOIUrl":null,"url":null,"abstract":"<div><div>In the minimum eigenvalue problem, we are given a collection of vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and the goal is to pick a subset <em>B</em> to maximize the minimum eigenvalue of the matrix <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>B</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⊤</mo></mrow></msubsup></math></span>. We give a <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>/</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup><mo>)</mo></mrow></math></span>-time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo></math></span> times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.</div></div>","PeriodicalId":54682,"journal":{"name":"Operations Research Letters","volume":"57 ","pages":"Article 107186"},"PeriodicalIF":0.8000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fast algorithms for maximizing the minimum eigenvalue in fixed dimension\",\"authors\":\"Adam Brown , Aditi Laddha , Mohit Singh\",\"doi\":\"10.1016/j.orl.2024.107186\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In the minimum eigenvalue problem, we are given a collection of vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and the goal is to pick a subset <em>B</em> to maximize the minimum eigenvalue of the matrix <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>B</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⊤</mo></mrow></msubsup></math></span>. We give a <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>/</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup><mo>)</mo></mrow></math></span>-time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo></math></span> times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.</div></div>\",\"PeriodicalId\":54682,\"journal\":{\"name\":\"Operations Research Letters\",\"volume\":\"57 \",\"pages\":\"Article 107186\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Operations Research Letters\",\"FirstCategoryId\":\"91\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0167637724001226\",\"RegionNum\":4,\"RegionCategory\":\"管理学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"OPERATIONS RESEARCH & MANAGEMENT SCIENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Operations Research Letters","FirstCategoryId":"91","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167637724001226","RegionNum":4,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
Fast algorithms for maximizing the minimum eigenvalue in fixed dimension
In the minimum eigenvalue problem, we are given a collection of vectors in , and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix . We give a -time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.
期刊介绍：
Operations Research Letters is committed to the rapid review and fast publication of short articles on all aspects of operations research and analytics. Apart from a limitation to eight journal pages, quality, originality, relevance and clarity are the only criteria for selecting the papers to be published. ORL covers the broad field of optimization, stochastic models and game theory. Specific areas of interest include networks, routing, location, queueing, scheduling, inventory, reliability, and financial engineering. We wish to explore interfaces with other fields such as life sciences and health care, artificial intelligence and machine learning, energy distribution, and computational social sciences and humanities. Our traditional strength is in methodology, including theory, modelling, algorithms and computational studies. We also welcome novel applications and concise literature reviews.
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