{"title":"均匀容能设施选址问题的lp逼近","authors":"Sapna Grover , Neelima Gupta , Samir Khuller","doi":"10.1016/j.disopt.2022.100723","DOIUrl":null,"url":null,"abstract":"<div><p><span><span>In this paper, we study uniform hard capacitated facility location problem. The standard LP for the problem is known to have an unbounded integrality gap. We present constant factor approximation by </span>rounding a solution to the standard LP with a slight </span><span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow></math></span> violation in the capacities.</p><p>Our result shows that the standard LP is not too bad.</p><p>Our algorithm is simple and more efficient as compared to the strengthened LP-based true approximation that uses the inefficient ellipsoid method with a separation oracle. True approximations are also known for the problem using local search techniques that suffer from the problem of convergence. Moreover, solutions based on standard LP are easier to integrate with other LP-based algorithms.</p><p>The result is also extended to give the first approximation for uniform hard capacitated <span><math><mi>k</mi></math></span>-facility location problem violating the capacities by a factor of <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow></math></span> and breaking the barrier of 2 in capacity violation. The result violates the cardinality by a factor of <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow></mfrac></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"LP-based approximation for uniform capacitated facility location problem\",\"authors\":\"Sapna Grover , Neelima Gupta , Samir Khuller\",\"doi\":\"10.1016/j.disopt.2022.100723\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span><span>In this paper, we study uniform hard capacitated facility location problem. The standard LP for the problem is known to have an unbounded integrality gap. We present constant factor approximation by </span>rounding a solution to the standard LP with a slight </span><span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow></math></span> violation in the capacities.</p><p>Our result shows that the standard LP is not too bad.</p><p>Our algorithm is simple and more efficient as compared to the strengthened LP-based true approximation that uses the inefficient ellipsoid method with a separation oracle. True approximations are also known for the problem using local search techniques that suffer from the problem of convergence. Moreover, solutions based on standard LP are easier to integrate with other LP-based algorithms.</p><p>The result is also extended to give the first approximation for uniform hard capacitated <span><math><mi>k</mi></math></span>-facility location problem violating the capacities by a factor of <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow></math></span> and breaking the barrier of 2 in capacity violation. The result violates the cardinality by a factor of <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow></mfrac></math></span>.</p></div>\",\"PeriodicalId\":50571,\"journal\":{\"name\":\"Discrete Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1572528622000330\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528622000330","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
LP-based approximation for uniform capacitated facility location problem
In this paper, we study uniform hard capacitated facility location problem. The standard LP for the problem is known to have an unbounded integrality gap. We present constant factor approximation by rounding a solution to the standard LP with a slight violation in the capacities.
Our result shows that the standard LP is not too bad.
Our algorithm is simple and more efficient as compared to the strengthened LP-based true approximation that uses the inefficient ellipsoid method with a separation oracle. True approximations are also known for the problem using local search techniques that suffer from the problem of convergence. Moreover, solutions based on standard LP are easier to integrate with other LP-based algorithms.
The result is also extended to give the first approximation for uniform hard capacitated -facility location problem violating the capacities by a factor of and breaking the barrier of 2 in capacity violation. The result violates the cardinality by a factor of .
期刊介绍:
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.