{"title":"Reducing the Chvátal rank through binarization","authors":"Gérard Cornuéjols, Vrishabh Patil","doi":"10.1016/j.orl.2024.107119","DOIUrl":null,"url":null,"abstract":"<div><p>In a classical paper, Chvátal introduced a rounding procedure for strengthening the polyhedral relaxation <em>P</em> of an integer program; applied recursively, the number of iterations needed to obtain the convex hull of the integer solutions in <em>P</em> is known as the Chvátal rank. Chvátal showed that this rank can be exponential in the input size <em>L</em> needed to describe <em>P</em>. We give a compact extended formulation of <em>P</em>, described by introducing binary variables, whose rank is polynomial in <em>L</em>.</p></div>","PeriodicalId":54682,"journal":{"name":"Operations Research Letters","volume":"54 ","pages":"Article 107119"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167637724000555/pdfft?md5=31547e69830dd0795992e92a3c7a822a&pid=1-s2.0-S0167637724000555-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Operations Research Letters","FirstCategoryId":"91","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167637724000555","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
In a classical paper, Chvátal introduced a rounding procedure for strengthening the polyhedral relaxation P of an integer program; applied recursively, the number of iterations needed to obtain the convex hull of the integer solutions in P is known as the Chvátal rank. Chvátal showed that this rank can be exponential in the input size L needed to describe P. We give a compact extended formulation of P, described by introducing binary variables, whose rank is polynomial in L.
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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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