AF: Small: Sublinear Algorithms for Graph Optimization Problems

AF：小：图优化问题的次线性算法

• 批准号: 1617851
• 负责人: Sanjeev Khanna
• 金额: $ 45万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1617851&HistoricalAwards=false
• 关键词: AF; Small; Sublinear; Algorithms; Graph

Very-large scale graphs routinely arise in applications where the data describes pairwise relationships among a set of objects. Some standard examples include the Web graph, social networks, and biological networks. The prevalence of such large data sets has led to a rapidly growing interest in the design of sublinear algorithms, that is, algorithms whose computational resource requirements are substantially smaller than the input size. As vast amounts of networked data is being collected and processed in diverse application domains, sublinear algorithms that accurately compute and describe relevant properties of the data will increasingly play an important role in computing on such data. The goal of this project is to develop sublinear algorithms for several fundamental graph optimization problems. The specific graph problems studied in this project are of both theoretical and practical interest, and are among the most well-studied problems in combinatorial optimization. Additionally, a study of these problems through the lens of sublinear algorithms is likely to yield new insights into computational aspects of these fundamental problems. The research proposed here will go hand-in-hand with educational and student-training initiatives, including mentoring and training of undergraduate and graduate students, and teaching in programs that introduce high-school students to exciting ideas in theoretical computer science.The research focus of this project is broadly divided into three parts. In the first part, the PIs consider streaming algorithms for graph problems where an input graph is revealed as a sequence of edge insertions and deletions. Some representative problems studied in this part include matching and cut problems in graphs. While both cuts and matchings have received considerable attention in the streaming literature, several important questions concerning their computability in the streaming model remain unresolved. In the second part, they consider communication-efficient protocols in a distributed setting when the input graph is partitioned across multiple sites. This model offers a natural abstraction for distributed computation and is closely related to the streaming model. A representative problem here is to understand the communication complexity of the maximum matching problem. The third part of this project investigates a new model for sketching graphs that consist of a (large) static part and a (small) dynamic part. The goal here is to understand if there exist compact sketches for several fundamental graph problems whose size is proportional to the size of the dynamic part of the input graph such that any updates to the dynamic part can be applied directly to the sketch.

在数据描述一组对象之间的成对关系的应用中，通常会出现非常大的比例图。一些标准示例包括网络图，社交网络和生物网络。如此大的数据集的流行率导致人们对sublinear算法设计的迅速增长，即，其计算资源需求大大小于输入大小的算法。随着大量的网络数据正在收集和处理各种应用域中，精确计算和描述数据相关属性的透明性算法将在计算此类数据中越来越重要。该项目的目的是为几种基本图优化问题开发均方根算法。该项目中研究的特定图形问题既具有理论上的兴趣，又是合并优化中最良好的问题之一。此外，通过透明性算法对这些问题的研究可能会产生对这些基本问题计算方面的新见解。这里提出的研究将与教育和学生培训计划并驾齐驱，包括对本科生和研究生的指导和培训，以及在课程中进行教学，这些计划向高中生介绍了理论计算机科学中令人兴奋的想法。该项目的研究重点大致分为三个部分。在第一部分中，PI考虑了用于图形问题的流算法，其中将输入图显示为一系列边缘插入和删除。在这部分中研究的一些代表性问题包括图表中的匹配和剪切问题。尽管剪辑和匹配在流文献中都受到了很大的关注，但有关其在流模型中计算性的一些重要问题仍未解决。在第二部分中，他们考虑在分布式设置中进行沟通效率的协议，当输入图跨多个站点分配时。该模型为分布式计算提供了自然的抽象，并且与流模型密切相关。这里的一个代表性问题是了解最大匹配问题的通信复杂性。该项目的第三部分研究了一个新模型，用于草图图形，该模型由（大）静态部分和（小）动态部分组成。这里的目的是了解几个基本图问题是否存在紧凑的草图，其大小与输入图的动态部分的大小成正比，以便可以直接应用于草图的动态零件的任何更新。

项目成果

Near-linear Size Hypergraph Cut Sparsifiers

• DOI: 10.1109/focs46700.2020.00015
• 发表时间: 2020-09
• 期刊: 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
• 作者: Yu Chen;S. Khanna;Ansh Nagda

Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

• DOI: 10.4230/lipics.icalp.2021.53
• 发表时间: 2021-06
• 作者: Yu Chen;S. Khanna;Ansh Nagda