{"title":"Multiplicative auction algorithm for approximate maximum weight bipartite matching","authors":"","doi":"10.1007/s10107-024-02066-3","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We present an <em>auction algorithm</em> using multiplicative instead of constant weight updates to compute a <span> <span>\\((1-\\varepsilon )\\)</span> </span>-approximate maximum weight matching (MWM) in a bipartite graph with <em>n</em> vertices and <em>m</em> edges in time <span> <span>\\(O(m\\varepsilon ^{-1})\\)</span> </span>, beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM ’14] that runs in <span> <span>\\(O(m\\varepsilon ^{-1}\\log \\varepsilon ^{-1})\\)</span> </span>. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a <span> <span>\\((1-\\varepsilon )\\)</span> </span>-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is <span> <span>\\(O(m\\varepsilon ^{-1})\\)</span> </span>, where <em>m</em> is the sum of the number of initially existing and inserted edges.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"55 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Programming","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10107-024-02066-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
We present an auction algorithm using multiplicative instead of constant weight updates to compute a \((1-\varepsilon )\)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time \(O(m\varepsilon ^{-1})\), beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM ’14] that runs in \(O(m\varepsilon ^{-1}\log \varepsilon ^{-1})\). Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a \((1-\varepsilon )\)-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is \(O(m\varepsilon ^{-1})\), where m is the sum of the number of initially existing and inserted edges.
期刊介绍：
Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.
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