{"title":"Updated Estimates for Algorithms for Packing\n2-Bar Charts in a Strip","authors":"S. A. Nazarenko","doi":"10.1134/S1990478924040112","DOIUrl":null,"url":null,"abstract":"<p> We consider a two-bar charts packing problem in which it is necessary to pack bar charts\nconsisting of two bars in a unit-height strip of minimum length. Each bar has a height of at most\n1 and unit length. The problem under consideration is NP-hard and generalizes the bin packing\nproblem and two-dimensional vector packing problem. This paper proves updated accuracy\nestimates and time complexity for several previously developed polynomial approximation\nalgorithms for the two-bar charts packing problem and particular cases of the problem. We show\nthe attainability of the estimates. Furthermore, we consider a problem of packing an unlimited\nnumber of bar charts belonging to\n<span>\\( k \\)</span> different types and propose a polynomial algorithm to solve the problem in\ncase\n<span>\\( k = \\text {const} \\)</span>.\n</p>","PeriodicalId":607,"journal":{"name":"Journal of Applied and Industrial Mathematics","volume":"18 4","pages":"759 - 774"},"PeriodicalIF":0.5800,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied and Industrial Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1134/S1990478924040112","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Engineering","Score":null,"Total":0}
引用次数: 0
Abstract
We consider a two-bar charts packing problem in which it is necessary to pack bar charts
consisting of two bars in a unit-height strip of minimum length. Each bar has a height of at most
1 and unit length. The problem under consideration is NP-hard and generalizes the bin packing
problem and two-dimensional vector packing problem. This paper proves updated accuracy
estimates and time complexity for several previously developed polynomial approximation
algorithms for the two-bar charts packing problem and particular cases of the problem. We show
the attainability of the estimates. Furthermore, we consider a problem of packing an unlimited
number of bar charts belonging to
\( k \) different types and propose a polynomial algorithm to solve the problem in
case
\( k = \text {const} \).
我们考虑一个双柱状图的包装问题，在这个问题中，有必要将由两个柱状图组成的柱状图包装在最小长度的单位高度条中。每个杆的高度最多为1，长度为单位。所考虑的问题是NP-hard问题，它推广了垃圾箱包装问题和二维向量包装问题。本文证明了几种先前开发的多项式近似算法对双柱状图包装问题的精度估计和时间复杂度，以及该问题的特殊情况。我们展示了估计的可实现性。此外，我们考虑了一个将无限数量的属于\( k \)不同类型的条形图打包的问题，并提出了一个多项式算法来解决\( k = \text {const} \)情况下的问题。
期刊介绍：
Journal of Applied and Industrial Mathematics is a journal that publishes original and review articles containing theoretical results and those of interest for applications in various branches of industry. The journal topics include the qualitative theory of differential equations in application to mechanics, physics, chemistry, biology, technical and natural processes; mathematical modeling in mechanics, physics, engineering, chemistry, biology, ecology, medicine, etc.; control theory; discrete optimization; discrete structures and extremum problems; combinatorics; control and reliability of discrete circuits; mathematical programming; mathematical models and methods for making optimal decisions; models of theory of scheduling, location and replacement of equipment; modeling the control processes; development and analysis of algorithms; synthesis and complexity of control systems; automata theory; graph theory; game theory and its applications; coding theory; scheduling theory; and theory of circuits.
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