{"title":"Max cut and semidefinite rank","authors":"Renee Mirka, David P. Williamson","doi":"10.1016/j.orl.2024.107067","DOIUrl":null,"url":null,"abstract":"<div><p><span>This paper considers the relationship between semidefinite programs (SDPs), matrix rank, and maximum cuts of graphs. Utilizing complementary slackness conditions for SDPs, we investigate when the rank 1 feasible solution corresponding to a max cut is the unique optimal solution to the Goemans-Williamson max cut SDP by showing the existence of an optimal dual solution with rank </span><span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span><span>. Our results consider connected bipartite graphs and graphs with multiple max cuts. We conclude with a conjecture for general graphs.</span></p></div>","PeriodicalId":54682,"journal":{"name":"Operations Research Letters","volume":"53 ","pages":"Article 107067"},"PeriodicalIF":0.8000,"publicationDate":"2024-01-17","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/S0167637724000038","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}
引用次数: 0
Abstract
This paper considers the relationship between semidefinite programs (SDPs), matrix rank, and maximum cuts of graphs. Utilizing complementary slackness conditions for SDPs, we investigate when the rank 1 feasible solution corresponding to a max cut is the unique optimal solution to the Goemans-Williamson max cut SDP by showing the existence of an optimal dual solution with rank . Our results consider connected bipartite graphs and graphs with multiple max cuts. We conclude with a conjecture for general graphs.
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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