{"title":"Iterated tour partitioning for Euclidean capacitated vehicle routing","authors":"Claire Mathieu, Hang Zhou","doi":"10.1002/rsa.21130","DOIUrl":null,"url":null,"abstract":"We give a probabilistic analysis of the unit‐demand Euclidean capacitated vehicle routing problem in the random setting. The objective is to visit all customers using a set of routes of minimum total length, such that each route visits at most k$$ k $$ customers. The best known polynomial‐time approximation is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the solution obtained by the ITP algorithm is arbitrarily close to the optimum when k$$ k $$ is either o(n)$$ o\\left(\\sqrt{n}\\right) $$ or ω(n)$$ \\omega \\left(\\sqrt{n}\\right) $$ , and they asked whether the ITP algorithm was “also effective in the intermediate range”. In this work, we show that the ITP algorithm is at best a (1+c0)$$ \\left(1+{c}_0\\right) $$ ‐approximation, for some positive constant c0$$ {c}_0 $$ , and is at worst a 1.915‐approximation.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"93 1","pages":"1056 - 1075"},"PeriodicalIF":0.9000,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/rsa.21130","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 1
Abstract
We give a probabilistic analysis of the unit‐demand Euclidean capacitated vehicle routing problem in the random setting. The objective is to visit all customers using a set of routes of minimum total length, such that each route visits at most k$$ k $$ customers. The best known polynomial‐time approximation is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the solution obtained by the ITP algorithm is arbitrarily close to the optimum when k$$ k $$ is either o(n)$$ o\left(\sqrt{n}\right) $$ or ω(n)$$ \omega \left(\sqrt{n}\right) $$ , and they asked whether the ITP algorithm was “also effective in the intermediate range”. In this work, we show that the ITP algorithm is at best a (1+c0)$$ \left(1+{c}_0\right) $$ ‐approximation, for some positive constant c0$$ {c}_0 $$ , and is at worst a 1.915‐approximation.
期刊介绍:
It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness.
Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.