Pub Date : 2018-05-09DOI: 10.1201/9781420010749.ch42
Weili Wu, D. Du
{"title":"Approximations for Steiner Minimum Trees","authors":"Weili Wu, D. Du","doi":"10.1201/9781420010749.ch42","DOIUrl":"https://doi.org/10.1201/9781420010749.ch42","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116972892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781420010749.ch35
L. Epstein, R. V. Stee
{"title":"Multidimensional Packing Problems","authors":"L. Epstein, R. V. Stee","doi":"10.1201/9781420010749.ch35","DOIUrl":"https://doi.org/10.1201/9781420010749.ch35","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126127290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781420010749.pt1
T. Gonzalez
{"title":"Introduction, Overview, and Notation","authors":"T. Gonzalez","doi":"10.1201/9781420010749.pt1","DOIUrl":"https://doi.org/10.1201/9781420010749.pt1","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123982339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Π be an NP-hard optimization problem, and let A be an approximation algorithm for Π. For an instance I of Π, let A(I) denote the objective value when running A on I, and let OPT (I) denote the optimal objective value. The approximation ratio of A for the instance I is RA(I) = A(I)/OPT (I), thus, when Π is minimization (maximization) problem RA(I) ≥ 1 (RA(I) ≤ 1). A polynomial time approximation scheme is an algorithm which takes as input an additional parameter, e, which determines the desired approximation ratio. This ratio can be arbitrarily close to 1, when e approaches 0. The time complexity of the scheme is polynomial in the input size but may be exponential in 1/e. This gives a clear trade-off between running time and quality of approximation. Formally,
{"title":"Polynomial Time Approximation Schemes","authors":"H. Shachnai, Tami Tamir","doi":"10.1201/9781351236423-8","DOIUrl":"https://doi.org/10.1201/9781351236423-8","url":null,"abstract":"Let Π be an NP-hard optimization problem, and let A be an approximation algorithm for Π. For an instance I of Π, let A(I) denote the objective value when running A on I, and let OPT (I) denote the optimal objective value. The approximation ratio of A for the instance I is RA(I) = A(I)/OPT (I), thus, when Π is minimization (maximization) problem RA(I) ≥ 1 (RA(I) ≤ 1). A polynomial time approximation scheme is an algorithm which takes as input an additional parameter, e, which determines the desired approximation ratio. This ratio can be arbitrarily close to 1, when e approaches 0. The time complexity of the scheme is polynomial in the input size but may be exponential in 1/e. This gives a clear trade-off between running time and quality of approximation. Formally,","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125861294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781420010749.ch48
T. Ibaraki, M. Yagiura
{"title":"Generalized Assignment Problem","authors":"T. Ibaraki, M. Yagiura","doi":"10.1201/9781420010749.ch48","DOIUrl":"https://doi.org/10.1201/9781420010749.ch48","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126731113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781351236423-32
S. Imahori, M. Yagiura, H. Nagamochi
{"title":"Practical Algorithms for Two-Dimensional Packing of Rectangles","authors":"S. Imahori, M. Yagiura, H. Nagamochi","doi":"10.1201/9781351236423-32","DOIUrl":"https://doi.org/10.1201/9781351236423-32","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122995510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781351236423-42
Niv Buchbinder, Moran Feldman
In this chapter we study fundamental results on maximizing a special class of functions called submodular functions under various combinatorial constraints. The study of submodular functions is motivated both by their many real world applications and by their frequent occurrence in more theoretical fields such as economy and algorithmic game theory. In particular, submodular functions and submodular maximization play a major role in combinatorial optimization as several well known combinatorial functions turn out to be submodular. A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. We discuss some of these examples further in the following. Let us begin by providing basic notation used throughout the chapter. We then give two definitions of submodular functions and prove that they are equivalent. Let N = {u1, u2, . . . , un} be a ground set of elements. For a set A and an element u ∈ N we denote the union A ∪ {u} by A+ u. Similarly, we denote A {u} as A− u. The following is the first definition of submodular functions.
{"title":"Submodular Functions Maximization Problems","authors":"Niv Buchbinder, Moran Feldman","doi":"10.1201/9781351236423-42","DOIUrl":"https://doi.org/10.1201/9781351236423-42","url":null,"abstract":"In this chapter we study fundamental results on maximizing a special class of functions called submodular functions under various combinatorial constraints. The study of submodular functions is motivated both by their many real world applications and by their frequent occurrence in more theoretical fields such as economy and algorithmic game theory. In particular, submodular functions and submodular maximization play a major role in combinatorial optimization as several well known combinatorial functions turn out to be submodular. A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. We discuss some of these examples further in the following. Let us begin by providing basic notation used throughout the chapter. We then give two definitions of submodular functions and prove that they are equivalent. Let N = {u1, u2, . . . , un} be a ground set of elements. For a set A and an element u ∈ N we denote the union A ∪ {u} by A+ u. Similarly, we denote A {u} as A− u. The following is the first definition of submodular functions.","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132244279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781351236423-25
Hans-Joachim Böckenhauer, J. Hromkovic, D. Komm
{"title":"Reoptimization of Hard Optimization Problems","authors":"Hans-Joachim Böckenhauer, J. Hromkovic, D. Komm","doi":"10.1201/9781351236423-25","DOIUrl":"https://doi.org/10.1201/9781351236423-25","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128888072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781351236423-35
P. Cappello, C. Coakley
{"title":"A Development and Deployment Framework for Distributed Branch-and-Bound","authors":"P. Cappello, C. Coakley","doi":"10.1201/9781351236423-35","DOIUrl":"https://doi.org/10.1201/9781351236423-35","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124362061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-05-09DOI: 10.1201/9781351236423-41
C. S. Sakuraba, M. Yagiura
{"title":"Linear Ordering Problem","authors":"C. S. Sakuraba, M. Yagiura","doi":"10.1201/9781351236423-41","DOIUrl":"https://doi.org/10.1201/9781351236423-41","url":null,"abstract":"","PeriodicalId":262519,"journal":{"name":"Handbook of Approximation Algorithms and Metaheuristics","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133780349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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