Approximation algorithms for Graph Drawing

图形绘制的近似算法

• 批准号: RGPIN-2015-06216
• 负责人: Biedl, Therese
• 金额: $ 2.62万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614012
• 关键词: Approximation; algorithms; Graph; Drawing

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.

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。可视化这样的图表有助于分析它们的结构，例如，检测集群或相似性。这就引发了图表绘制的问题：开发出自动将图表转换为易于阅读的图片的算法。 用户研究表明，图表易读性的最重要标准是将交叉次数降至最低。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。有许多绘制平面图的算法，但尽管它们在某些质量指标方面被证明是好的，但它们往往给出的图片远不是给定图形的最佳可能。 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性能保证且可证明在最优因子范围内的算法。目前，用于绘制平面图的近似算法很少。即使是使用著名的近似算法技术(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，以最小化面积为目标，并计划将工作扩展到开发最小面积图的近似算法。这方面的一个主要挑战是开发新的平面图绘制下界工具，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，所以我提议的研究的另一条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被用于应用程序中，为它们开发近似算法是一个令人兴奋的挑战。这样的算法将给出可证明距离给定输入图形的最佳可能绘制不太远的图形绘制，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。