{"title":"Combinatorial optimization with rational objective functions","authors":"N. Megiddo","doi":"10.1145/800133.804326","DOIUrl":null,"url":null,"abstract":"Let A be the problem of minimizing c1x1+...+cnxn subject to certain constraints on x=(x1,...,xn), and let B be the problem of minimizing (a0+a1x1+...+anxn)/(b0+b1x1+...+bnxn) subject to the same constraints, assuming the denominator is always positive. It is shown that if A is solvable within O[p(n)] comparisons and O[q(n)] additions, then B is solvable in time O[p(n)(q(n)+p(n))]. This applies to most of the “network” algorithms. Consequently, minimum ratio cycles, minimum ratio spanning trees, minimum ratio (simple) paths, maximum ratio weighted matchings, etc., can be computed within polynomial-time in the number of variables. This improves a result of E. L. Lawler, namely, that a minimum ratio cycle can be computed within a time bound which is polynomial in the number of bits required to specify an instance of the problem. A recent result on minimum ratio spanning trees by R. Chandrasekaran is also improved by the general arguments presented in this paper. Algorithms of time-complexity O(¦E¦ • ¦V¦2•log¦V¦) for a minimum ratio cycle and O(¦E¦ • log2¦V¦ • log log ¦V¦) for a minimum ratio spanning tree are developed.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"509","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the tenth annual ACM symposium on Theory of computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/800133.804326","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 509
Abstract
Let A be the problem of minimizing c1x1+...+cnxn subject to certain constraints on x=(x1,...,xn), and let B be the problem of minimizing (a0+a1x1+...+anxn)/(b0+b1x1+...+bnxn) subject to the same constraints, assuming the denominator is always positive. It is shown that if A is solvable within O[p(n)] comparisons and O[q(n)] additions, then B is solvable in time O[p(n)(q(n)+p(n))]. This applies to most of the “network” algorithms. Consequently, minimum ratio cycles, minimum ratio spanning trees, minimum ratio (simple) paths, maximum ratio weighted matchings, etc., can be computed within polynomial-time in the number of variables. This improves a result of E. L. Lawler, namely, that a minimum ratio cycle can be computed within a time bound which is polynomial in the number of bits required to specify an instance of the problem. A recent result on minimum ratio spanning trees by R. Chandrasekaran is also improved by the general arguments presented in this paper. Algorithms of time-complexity O(¦E¦ • ¦V¦2•log¦V¦) for a minimum ratio cycle and O(¦E¦ • log2¦V¦ • log log ¦V¦) for a minimum ratio spanning tree are developed.
设A是最小化c1x1+…的问题+cnxn在x=(x1,…,xn)上有一定约束,设B为在相同约束下最小化(a0+a1x1+…+anxn)/(b0+b1x1+…+bnxn)的问题,假设分母总是正的。证明了如果A在O[p(n)]个比较和O[q(n)]个加法内可解,那么B在O[p(n))(q(n)+p(n)))时间内可解。这适用于大多数“网络”算法。因此,最小比值循环、最小比值生成树、最小比值(简单)路径、最大比值加权匹配等,都可以在变量数量的多项式时间内计算出来。这改进了E. L. Lawler的一个结果,即可以在指定问题实例所需的比特数的多项式的时间范围内计算最小比率周期。本文提出的一般论点也改进了R. Chandrasekaran关于最小比值生成树的最新结果。给出了最小比值生成树的时间复杂度为O(……V…2·log…)和O(……log2·V…·log log…)的算法。