Approximation algorithms for Graph Drawing

在社交网络、街道地图、数据库的实体关系图、组织层次结构和软件工程的控制流程图等应用程序中，底层结构可以抽象地描述为图：它由顶点(人、站点、实体、员工、功能)和它们之间的边(关系、街道、函数调用)组成。可视化这类图形有助于分析它们的结构，例如用于检测簇或相似性。这引发了图形绘制的问题：开发算法自动将图形转换为易于阅读的图片。*用户研究表明，图形绘制的易读性最重要的标准是尽量减少交叉的数量。因此，图形绘制的一个重要分支是如何绘制平面图，即根本不需要任何交叉点就可以绘制的图形。绘制平面图的算法有很多，但虽然它们在某些质量指标上是被证明是好的，但是对于给定的图，它们所给出的图往往远不是最好的。*所提出的研究主题是平面图绘制的近似算法，即开发对所有图都具有可证明的性能保证的算法。目前，用于绘制平面图的近似算法很少。即使是使用众所周知的近似算法技术(如LP松弛和贝克技术)的关键成分也很少被研究。我最近为了最小化面积的目标在这些关键成分上取得了进展，并计划将工作扩展到开发用于最小面积绘图的近似算法。这方面的一个主要挑战是开发用于平面图绘制的新的下界工具，因为现有的下界工具已被证明不足以用于近似算法。可能不存在近似算法。因此，我提出的研究的另一个途径是通过应用复杂性理论中的APX-硬度技术来证明这种不可能性。从长远来看，许多其他的图形绘制目标也在应用中，为它们开发近似算法是一个令人兴奋的挑战。这样的算法将给出与给定输入图形的最佳可能图形相差不太远的图形绘制，因此对于显示上述应用领域的图形结构来说应该是一个显著的改进。

In applications such as social networks, street maps, entity-relationship diagrams for data bases, organizational hierarchies and control flow diagrams for software engineering, the underlying structure can be described abstractly as a graph: it consists of vertices (humans, sites, entities, employees, functions) and edges (relationships, streets, function calls) between them. Visualizing such graphs helps in analyzing their structure, for example for detecting clusters or similarities.  This motivates the problem of graph drawing: develop algorithms that automatically convert a graph into a picture that is easy to read.******User studies have shown that the most important criteria for legibility of a graph drawing is to minimize the number of crossings. Hence an important sub-area of graph drawing is how to draw a planar graph, i.e., a graph that could be drawn without any crossings at all. Many algorithms for drawing planar graphs exist, but although they are provably good with regard to some quality measure, they often give pictures that are far from best-possible for a given graph.******The topic of the proposed research is approximation algorithms for planar graph drawing, i.e., to develop algorithms with performance guarantees that are provably within a factor of the optimum for all graphs. Currently very few approximation algorithms for drawing planar graphs exist. Even the key ingredients for using well-known techniques for approximation algorithms (such as LP relaxation and Baker's technique) have rarely been studied.  I recently made progress on these key ingredients for the objective of minimizing the area, and plan to expand the work into developing approximation algorithms for minimum-area drawings  A major challenge for this is to develop new lower bound tools for planar graph drawing, since the existing lower-bound tools have proved insufficient for approximation algorithms.  Possibly no approximation algorithms exist, and so another avenue of my proposed research is to prove such impossibilites by applying the techniques of APX-hardness from complexity theory.  ***In the longer term, many other graph drawing objectives are used in the applications, and developing approximation algorithms for them is an exciting challenge. Such algorithms would give graph drawings that are provably not too far from the best-possible drawing of the given input graph, and hence should be a significant improvement for displaying the graph-structures of the above application areas.**