Approximation algorithms for Graph Drawing

在诸如社交网络、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形：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。可视化这些图有助于分析它们的结构，例如用于检测聚类或相似性。这激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，图形绘制易读性的最重要标准是尽量减少交叉点的数量。因此，图形绘制的一个重要的子领域是如何绘制一个平面图形，即一个可以绘制没有任何交叉点的图形。存在许多绘制平面图形的算法，但是尽管它们在某些质量度量方面被证明是好的，但它们通常给出的图像远不是给定图形的最佳可能。******提出的研究主题是平面图形绘制的近似算法，即开发具有性能保证的算法，该算法可证明在所有图形的最优因子范围内。目前绘制平面图形的近似算法很少。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。我最近在这些关键成分上取得了进展，以实现最小化面积的目标，并计划将工作扩展到开发最小面积绘图的近似算法。这方面的主要挑战是开发新的平面图形绘图的下限工具，因为现有的下限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的apx硬度技术来证明这种不可能。***从长远来看，应用程序中使用了许多其他图形绘制目标，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的最佳可能图相差不远的图，因此对于显示上述应用领域的图结构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.**