AF: Small: Sublinear Algorithms for Graph Optimization Problems

超大规模图通常出现在数据描述一组对象之间的成对关系的应用中。一些标准的例子包括Web图、社交网络和生物网络。这种大数据集的流行导致了对次线性算法设计的兴趣迅速增长，次线性算法是指计算资源需求远小于输入大小的算法。随着大量的网络数据在不同的应用领域中被收集和处理，准确计算和描述数据的相关属性的次线性算法将在此类数据的计算中发挥越来越重要的作用。这个项目的目标是为几个基本的图优化问题开发次线性算法。在这个项目中研究的特定的图形问题的理论和实际利益，是在组合优化中最好的研究问题之一。此外，这些问题的研究，通过透镜的次线性算法可能会产生新的见解，这些基本问题的计算方面。本研究计划将与教育和学生培训计划齐头并进，包括指导和培训本科生和研究生，以及在课程中向高中生介绍理论计算机科学中令人兴奋的想法。本项目的研究重点大致分为三个部分。在第一部分中，PI考虑流算法的图形问题，其中输入图显示为一系列的边缘插入和删除。这一部分研究了图的匹配和割问题。虽然切割和匹配在流媒体文献中受到了相当大的关注，但关于它们在流媒体模型中的可计算性的几个重要问题仍然没有得到解决。在第二部分中，当输入图跨多个站点分区时，他们考虑分布式环境中的通信高效协议。该模型为分布式计算提供了一个自然的抽象，并且与流模型密切相关。这里的一个代表性问题是理解最大匹配问题的通信复杂性。本项目的第三部分研究了一种新的模型，用于绘制由（大）静态部分和（小）动态部分组成的图形。这里的目标是了解是否存在几个基本图问题的紧凑草图，这些问题的大小与输入图的动态部分的大小成比例，以便对动态部分的任何更新都可以直接应用于草图。

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.