Algorithms for near-planar graphs

地铁地图、实体关系图、社会网络、街道地图、层级组织、控制流程图和生化途径有什么共同之处？它们都是可以建模为图形的结构和关系的示例：由顶点（表示站点、节点、点……）和边(即顶点对（表示连接、弧线、街道、含义……）组成的抽象结构。。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？”，“有多少人彼此都认识？”，“从A到B的最短路径是什么？”）这些问题被翻译成图的语言，就变成了众所周知的着色、团和最短路径优化问题。图的一个常用子类是平面图，它可以在平面上绘制而不需要交叉。它们经常出现在应用程序中：例如，表示几何站点之间相互作用的图形（如Delauney三角形）通常是平面的。平面图中的优化问题已经被研究了几十年，并且已知的许多例子表明，在平面图中优化问题比在任意图中更有效地解决。也存在一些通用的工具来开发平面图形的算法，如贝克近似格式、分隔符和二维。在现实生活中，道路网络等应用程序产生的图形通常接近平面，但确实需要一些交叉点（例如，地下通道与桥梁相交的地方）。近年来，这引起了人们对所谓的“近平面图”的极大兴趣——这是一个包罗万象的术语，适用于任何一类不一定是平面的，但在某种程度上是“接近”的图。这些主要的例子是小格图（即，它们可以嵌入在小格的曲面中），k-平面图（即，每条边最多可以有k个交叉点）或几何图的概化（例如，高阶德劳尼三角剖分）。本研究的目标是为这类近平面图的优化问题找到更好的算法。对于小属图（一般来说，所有所谓的h次自由图），我们已经知道了很多。然而，对于其他近平面图，大多数现有结果要么关注属性（“有多少条边？”）或者是通过平面化图得到的。很少有算法是已知的优化问题；我建议探索这样的算法，短期目标是将许多已知的平面图结果转化为近平面图，长期目标是建立通用的工具来开发近平面图的算法。这个主题非常适合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。

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.