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

这个项目福尔斯计算复杂性的学科，它研究有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以这样的方式实现，即时间，内存空间和其他资源的量与输入大小适度缩放。后者包含了来自科学和工程各个分支的数千个计算问题，这些问题在某种意义上是等价的，即一个有效的算法意味着所有的算法。P vs NP问题询问是否存在有效的算法来解决这些问题。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世纪世纪重大挑战的七个千年奖问题之一。一个肯定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，它也将产生一种方法来打破目前正在使用的加密系统，事实上，这意味着不可能通过互联网进行安全通信。这个项目适合于解决这个基本而重要的问题。特别是，它建立了一个紧密的联系，在这个问题和数量的NP完全问题的实例可以有效地压缩，而不影响他们的可解性。如果P=NP，那么NP完全决策问题可以有效地压缩到一个单一的bit。另一方面，在一个比PNP更强的假设下，PI已经建立了NP完全问题，如可满足性和顶点覆盖不允许任何非平凡的压缩量。该方法取决于存在的高密度子集的整数没有算术级数的长度为3。 该项目进一步发展了这种方法，并研究了其对其他感兴趣的计算参数的影响。该项目还涉及一个系统的研究使用的高密度子集的整数没有算术级数的计算复杂性的一定长度，并开发新的应用程序。上述结构处理确定性压缩方案。出于密码学和其他原因，对随机化设置的扩展是有意义的。一种可能的办法是去随机化。在这种情况下，该项目研究了典型正确的去随机化的潜力，其中一个目标是有效的确定性模拟，这些模拟在大多数但不一定是所有给定长度的输入上都能正确运行。

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.