AF:Small: Applications of AP-free sets and derandomization

AF:Small：无 AP 集和去随机化的应用

• 批准号: 1017597
• 负责人: Dieter van Melkebeek
• 金额: $ 49.99万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1017597&HistoricalAwards=false
• 关键词: AF; Small; Applications; AP; free

This project falls within the discipline of computational complexity, which studies the power and limitations of efficient computation. The area develops models that represent the various capabilities of digital computing devices. It aims to determine which transformations can be realized in such a way that the amount of time, memory space, and other resources scale moderately with the input size.Of central importance is the class of so-called NP-complete problems. The latter contains thousands of computational problems from all branches of science and engineering that have been shown equivalent in the sense that an efficient algorithm for one implies such an algorithm for all. The P vs NP question asks whether efficient algorithms exist for these problems. It constitutes the main open question in theory of computing and is one of the seven millennium prize problems proposed by the Clay Mathematics Institute as grand challenges for the 21st century. A positive answer would open up tremendous possibilities that would affect most human endeavors. On the other hand, it would also yield a way to break the cryptographic systems that are currently in use and, in fact, imply the impossibility of secure communication over the internet.This project fits into the quest to settle that fundamental and important problem. In particular, it establishes a tight connection between that question and the amount by which instances of NP-complete problems can be efficiently compressed without affecting their solvability.If P=NP, then NP-complete decision problems can be efficiently compressed to a single bit. On the other hand, under a hypothesis that is somewhat stronger than PNP, the PI has established that NP-complete problems like satisfiability and vertex cover do not allow any nontrivial amount of compression. The approach hinges on the existence of high-density subsets of the integers without arithmetic progressions of length 3. This project further develops that approach and investigates its implications for other computational parameters of interest. The project also involves a systematic study of the use of high-density subsets of the integers without arithmetic progressions of certain lengths in computational complexity, and the development of new applications.The above construction handles deterministic compression schemes. For cryptographic and other reasons the extension to the randomized setting is of interest. One possible approach involves derandomization. In this context the project investigates the potential of typically-correct derandomization, where one aims for efficient deterministic simulations that behave correctly on most but not necessarily all inputs of any given length.

该项目属于计算复杂性学科，研究高效计算的能力和局限性。该领域开发代表数字计算设备各种功能的模型。它的目的是确定哪些转换可以通过时间、内存空间和其他资源随输入大小适度缩放的方式实现。最重要的是所谓的 NP 完全问题。后者包含来自科学和工程各个分支的数千个计算问题，这些问题已被证明是等效的，即对一个人有效的算法意味着对所有人都有这样的算法。 P 与 NP 问题询问是否存在针对这些问题的有效算法。它构成了计算理论中的主要开放问题，也是克莱数学研究所提出的 21 世纪重大挑战的七个千年奖问题之一。一个积极的答案将开启巨大的可能性，影响大多数人类的努力。另一方面，它也将产生一种破解当前使用的加密系统的方法，事实上，这意味着通过互联网进行安全通信是不可能的。该项目适合解决这一基本且重要的问题。特别是，它在该问题与 NP 完全问题的实例可以有效压缩而不影响其可解性的情况之间建立了紧密的联系。如果 P=NP，则 NP 完全决策问题可以有效地压缩为一位。另一方面，在比 PNP 更强的假设下，PI 已确定 NP 完全问题（如可满足性和顶点覆盖）不允许任何不平凡的压缩量。该方法取决于是否存在长度为 3 的算术级数的高密度整数子集。该项目进一步开发了该方法并研究了其对其他感兴趣的计算参数的影响。该项目还涉及对计算复杂性上没有一定长度算术级数的整数高密度子集的使用的系统研究，以及新应用程序的开发。上述结构处理确定性压缩方案。出于密码学和其他原因，对随机设置的扩展很有意义。一种可能的方法涉及去随机化。在这种背景下，该项目研究了典型正确的去随机化的潜力，其目标是高效的确定性模拟，该模拟在大多数但不一定是任何给定长度的所有输入上表现正确。