AF: Small: The Complexity of Random CSPs

这个项目关注的是理解解决被称为“约束满足问题”的非常大的随机生成任务的计算难度。这项研究的一个主要动机是密码学。用于传输和计算安全加密数据的加密协议依赖于我们轻松生成随机、难以解决的计算任务的能力。例如，许多加密协议依赖于这样的假设：如果将两个随机素数相乘，那么计算上很难分解结果的乘积。最近的密码学研究一直在寻找易于生成的难题的新来源，以增加灵活性和促进不同类型的密码学任务。然而，尽管我们对哪种类型的计算任务在最坏的情况下很难解决有了相当好的理解，但是当我们随机选择哪种类型的计算任务时，我们对它们的难度却知之甚少。“约束满足问题（csp）”这类问题似乎是一个很好的、潜在的有用的候选问题，因为这类问题很难随机选择。这个项目将涉及进一步加深我们对随机csp计算可行性的理解，研究所涉及的各种参数（如约束与变量的比例，约束的种类等）如何影响其易/难性。该项目的另一个方面是为卡耐基梅隆大学计算机科学本科生和研究生提供科学和教育培训，以及广泛传播研究成果。在更技术性的层面上，该项目有几条与随机csp的计算复杂性以及其解决方案的算法相关的查询线。PI将考虑约束密度、约束类型、近似/反驳算法的质量和运行时间之间的权衡。特别地，PI将研究在随机csp的背景下强大的“平方和（SOS）半确定规划层次”的能力和局限性。PI还将基于假设随机csp的难解性，研究非csp和学习理论问题的潜在新硬度结果。

This project is concerned with understanding the computational difficulty of solving very large, randomly generated tasks called "constraint satisfaction problems." One major motivation for this study is cryptography. Cryptographic protocols used to transmit and compute on securely encrypted data rely on our ability to easily generate random, hard-to-solve computational tasks. As an example, many cryptographic protocols rely on the assumption that if you multiply together two random prime numbers of many digits, it is computationally difficult to factor the resulting product. Recent research in cryptography has sought new sources of easy-to-generate hard problems, both for added flexibility and for facilitating different types of cryptographic tasks. However, although we have a fairly good understanding of what kinds of computational tasks can be hard to solve in the worst case, we do not know nearly as much about what kinds of computational tasks are expected to be hard when they are chosen at random. The class of "constraint satisfaction problems (CSPs)" seems to be a good and potentially useful candidate for a class of problems that is hard when chosen at random. This project will involve furthering our understanding of the computational feasibility of random CSPs, studying how the various parameters involved (such as the ratio of constraints to variables, the kinds of constraints, etc.) affects their easiness/difficulty. An additional aspect of the project will be scientific and educational training for computer science undergraduate and graduate students at Carnegie Mellon University, as well as wide dissemination of the research produced.At a more technical level, the project has several lines of inquiry related both to the computational complexity of random CSPs, as well as algorithms for their solution. The PI will consider the tradeoffs between constraint density, constraint type, quality of approximation/refutation algorithms, and running time. Particularly, the PI will investigate the power and limitations of the powerful "Sum-of-Squares (SOS) semidefinite programming hierarchy" in the context of random CSPs. The PI will also investigate potential new hardness results for non-CSPs and learning theory problems, based on the assumed intractability of random CSPs.