{"title":"双模整数线性规划的强多项式算法","authors":"S. Artmann, R. Weismantel, R. Zenklusen","doi":"10.1145/3055399.3055473","DOIUrl":null,"url":null,"abstract":"We present a strongly polynomial algorithm to solve integer programs of the form max{cT x: Ax≤ b, xεℤn }, for AεℤmXn with rank(A)=n, bε≤m, cε≤n, and where all determinants of (nXn)-sub-matrices of A are bounded by 2 in absolute value. In particular, this implies that integer programs max{cT x : Q x≤ b, xεℤ≥0n}, where Qε ℤmXn has the property that all subdeterminants are bounded by 2 in absolute value, can be solved in strongly polynomial time. We thus obtain an extension of the well-known result that integer programs with constraint matrices that are totally unimodular are solvable in strongly polynomial time.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"36 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"60","resultStr":"{\"title\":\"A strongly polynomial algorithm for bimodular integer linear programming\",\"authors\":\"S. Artmann, R. Weismantel, R. Zenklusen\",\"doi\":\"10.1145/3055399.3055473\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We present a strongly polynomial algorithm to solve integer programs of the form max{cT x: Ax≤ b, xεℤn }, for AεℤmXn with rank(A)=n, bε≤m, cε≤n, and where all determinants of (nXn)-sub-matrices of A are bounded by 2 in absolute value. In particular, this implies that integer programs max{cT x : Q x≤ b, xεℤ≥0n}, where Qε ℤmXn has the property that all subdeterminants are bounded by 2 in absolute value, can be solved in strongly polynomial time. We thus obtain an extension of the well-known result that integer programs with constraint matrices that are totally unimodular are solvable in strongly polynomial time.\",\"PeriodicalId\":20615,\"journal\":{\"name\":\"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing\",\"volume\":\"36 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-06-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"60\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3055399.3055473\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3055399.3055473","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A strongly polynomial algorithm for bimodular integer linear programming
We present a strongly polynomial algorithm to solve integer programs of the form max{cT x: Ax≤ b, xεℤn }, for AεℤmXn with rank(A)=n, bε≤m, cε≤n, and where all determinants of (nXn)-sub-matrices of A are bounded by 2 in absolute value. In particular, this implies that integer programs max{cT x : Q x≤ b, xεℤ≥0n}, where Qε ℤmXn has the property that all subdeterminants are bounded by 2 in absolute value, can be solved in strongly polynomial time. We thus obtain an extension of the well-known result that integer programs with constraint matrices that are totally unimodular are solvable in strongly polynomial time.