{"title":"广义正则网格上的填充问题:用整数线性规划的抽象层次","authors":"Hao Hua , Benjamin Dillenburger","doi":"10.1016/j.gmod.2023.101205","DOIUrl":null,"url":null,"abstract":"<div><p>Packing a designated set of shapes on a regular grid is an important class of operations research problems that has been intensively studied for more than six decades. Representing a <span><math><mi>d</mi></math></span>-dimensional discrete grid as <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we formalise the generalised regular grid (GRG) as a surjective function from <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> to a geometric tessellation in a physical space, for example, the cube coordinates of a hexagonal grid or a quasilattice. This study employs 0-1 integer linear programming (ILP) to formulate the polyomino tiling problem with adjacency constraints. Rotation & reflection invariance in adjacency are considered. We separate the formal ILP from the topology & geometry of various grids, such as Ammann-Beenker tiling, Penrose tiling and periodic hypercube. Based on cutting-edge solvers, we reveal an intuitive correspondence between the integer program (a pattern of algebraic rules) and the computer codes. Models of packing problems in the GRG have wide applications in production system, facility layout planning, and architectural design. Two applications in planning high-rise residential apartments are illustrated.</p></div>","PeriodicalId":55083,"journal":{"name":"Graphical Models","volume":"130 ","pages":"Article 101205"},"PeriodicalIF":2.5000,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Packing problems on generalised regular grid: Levels of abstraction using integer linear programming\",\"authors\":\"Hao Hua , Benjamin Dillenburger\",\"doi\":\"10.1016/j.gmod.2023.101205\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Packing a designated set of shapes on a regular grid is an important class of operations research problems that has been intensively studied for more than six decades. Representing a <span><math><mi>d</mi></math></span>-dimensional discrete grid as <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we formalise the generalised regular grid (GRG) as a surjective function from <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> to a geometric tessellation in a physical space, for example, the cube coordinates of a hexagonal grid or a quasilattice. This study employs 0-1 integer linear programming (ILP) to formulate the polyomino tiling problem with adjacency constraints. Rotation & reflection invariance in adjacency are considered. We separate the formal ILP from the topology & geometry of various grids, such as Ammann-Beenker tiling, Penrose tiling and periodic hypercube. Based on cutting-edge solvers, we reveal an intuitive correspondence between the integer program (a pattern of algebraic rules) and the computer codes. Models of packing problems in the GRG have wide applications in production system, facility layout planning, and architectural design. Two applications in planning high-rise residential apartments are illustrated.</p></div>\",\"PeriodicalId\":55083,\"journal\":{\"name\":\"Graphical Models\",\"volume\":\"130 \",\"pages\":\"Article 101205\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2023-10-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Graphical Models\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1524070323000358\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Graphical Models","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1524070323000358","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
Packing problems on generalised regular grid: Levels of abstraction using integer linear programming
Packing a designated set of shapes on a regular grid is an important class of operations research problems that has been intensively studied for more than six decades. Representing a -dimensional discrete grid as , we formalise the generalised regular grid (GRG) as a surjective function from to a geometric tessellation in a physical space, for example, the cube coordinates of a hexagonal grid or a quasilattice. This study employs 0-1 integer linear programming (ILP) to formulate the polyomino tiling problem with adjacency constraints. Rotation & reflection invariance in adjacency are considered. We separate the formal ILP from the topology & geometry of various grids, such as Ammann-Beenker tiling, Penrose tiling and periodic hypercube. Based on cutting-edge solvers, we reveal an intuitive correspondence between the integer program (a pattern of algebraic rules) and the computer codes. Models of packing problems in the GRG have wide applications in production system, facility layout planning, and architectural design. Two applications in planning high-rise residential apartments are illustrated.
期刊介绍：
Graphical Models is recognized internationally as a highly rated, top tier journal and is focused on the creation, geometric processing, animation, and visualization of graphical models and on their applications in engineering, science, culture, and entertainment. GMOD provides its readers with thoroughly reviewed and carefully selected papers that disseminate exciting innovations, that teach rigorous theoretical foundations, that propose robust and efficient solutions, or that describe ambitious systems or applications in a variety of topics.
We invite papers in five categories: research (contributions of novel theoretical or practical approaches or solutions), survey (opinionated views of the state-of-the-art and challenges in a specific topic), system (the architecture and implementation details of an innovative architecture for a complete system that supports model/animation design, acquisition, analysis, visualization?), application (description of a novel application of know techniques and evaluation of its impact), or lecture (an elegant and inspiring perspective on previously published results that clarifies them and teaches them in a new way).
GMOD offers its authors an accelerated review, feedback from experts in the field, immediate online publication of accepted papers, no restriction on color and length (when justified by the content) in the online version, and a broad promotion of published papers. A prestigious group of editors selected from among the premier international researchers in their fields oversees the review process.
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