Algorithms for near-planar graphs

近平面图的算法

• 批准号: RGPIN-2020-03958
• 负责人: Biedl, Therese
• 金额: $ 2.99万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739590
• 关键词: Algorithms; near; planar; graphs

What do a subway map, an entity-relationship diagram, a social network, a street map, a hierarchical organization, a control flow diagram and a biochemical pathway have in common? They are all examples of structures and relationships that can be modeled as a graph: an abstract structure consisting of vertices (representing sites, nodes, points, . . . ) and edges, i.e., pairs of vertices (representing connections, arcs, streets, implications, . . . ). Typically one wants to know some properties of these structures ("how many different frequencies do radio sites need to avoid interference?", "how many people all mutually know each other?", "what is the shortest way to get from A to B?") which, when translated to the language of graphs, become the well-known optimization problems of coloring, clique and shortest path. One commonly studied subclass of graphs are the planar graphs, which can be drawn in the plane without crossings. They frequently appear in applications: for example graphs (such as Delauney triangulations) that express interactions between geometric sites are often planar. Optimization-problems in planar graphs have been studied intensely for many decades, and many examples are known where optimization problems can be solved more efficiently in planar graphs than in arbitrary graphs. There also exist some general-purpose tools for developing algorithms for planar graphs, such as Baker's approximation scheme, separators and bidimensionality. In real life, graphs arising from applications such as road networks are often close to planar, but do require a few crossings (e.g. where an underpass meets a bridge). In recent years, this has caused much interest in so-called "near-planar graphs"---a catch-all term for any class of graphs that are not necessarily planar, but that are "close" in some way. Prime examples of these are graphs of small genus (i.e., they can be embedded in a surface of small genus), k-planar graphs (i.e., every edge can have at most k crossings) or generalization of geometric graphs (e.g., higher-order Delauney triangulation). The goal of the proposed research is to find better algorithms for optimization problems for such near-planar graphs. Quite a bit is known already for graphs of small genus (and generally, all so-called H-minor free graphs). However, for other near-planar graphs most existing results either focus on properties ("how many edges are there?") or are trivially obtained by planarizing the graph. Few algorithms are known for optimization problems; I propose to explore such algorithms, with the short-term objective of transferring numerous known results for planar graphs to near-planar graphs, and the long-term vision of establishing general-purpose tools for developing algorithms for near-planar graphs. This topic is ideally suited for HQP training, enabling them to develop the logical thinking and problem-solving skills crucial for their later work in the software industry or academia.

地铁地图、实体关系图、社交网络、街道地图、层级组织、控制流程图和生化路径有什么共同之处？它们都是可以建模为图的结构和关系的示例：由顶点(代表站点、节点、点)组成的抽象结构。。。)和边，即顶点对(表示连接、圆弧、街道、暗示、.。。)。通常，人们想知道这些结构的一些特性(“无线电站点需要多少不同的频率才能避免干扰？”，“有多少人相互认识？”，“从A到B的最短方式是什么？”)当这些问题被翻译成图的语言时，就变成了众所周知的着色、团和最短路径优化问题。平面图是一类通常被研究的图，它可以在平面上画出来而不需要交叉。它们经常出现在应用程序中：例如，表示几何位置之间交互的图(如Delauney三角剖分)通常是平面的。平面图中的优化问题已经被深入研究了几十年，许多例子表明，在平面图中可以比在任意图中更有效地解决优化问题。也有一些通用的工具来开发平面图的算法，如Baker的近似格式、分隔符和二维性。在现实生活中，道路网络等应用产生的图形通常接近于平面，但确实需要几个交叉点(例如，在地下通道与桥梁的交叉处)。近年来，这引起了人们对所谓的“近平面图”的极大兴趣-这是一个笼统的术语，指的是任何一类不一定是平面的，但在某种程度上是“接近”的图。这些图的主要例子是小亏格的图(即，它们可以嵌入小亏格的曲面中)、k平面图(即，每条边至多可以有k个交叉点)或几何图的推广(例如，高阶Delauney三角剖分)。这项研究的目的是为这类近平面图的优化问题找到更好的算法。对于小亏格的图(通常是所有所谓的H-次子自由图)，已经知道了相当多。然而，对于其他近平面图，大多数已有的结果要么集中在性质上(“有多少条边？”)或者是通过平面化图形来平凡地获得。很少有算法是已知的优化问题；我建议探索这样的算法，短期目标是将大量平面图的已知结果转移到近平面图，长期目标是建立通用工具来开发近平面图的算法。这一主题非常适合HQP培训，使他们能够发展逻辑思维和解决问题的技能，这对他们以后在软件行业或学术界的工作至关重要。