CAREER: Randomized Computations and Probabilistically Checkable Proofs

职业：随机计算和概率可检查证明

• 批准号: 9984703
• 负责人: Luca Trevisan
• 金额: $ 12.59万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9984703&HistoricalAwards=false
• 关键词: CAREER; Randomized; Computations; Probabilistically; Checkable

Summary:The use of randomness affects computations in dramatic and not yet fully understood ways: in algorithm design it yields simpler and more efficient ways to solve computational problems; in complexity theory it suggests new concepts and models that lead sometimes to unexpected (and far-reaching) results. This career development project involves a collection of research and educational activities related to computational randomness.The research component of this project deals with two main themes. One goal is the development of general tools that can be used to make randomized algorithms more robust, so that they can work even if they are implemented using biased, and limited, sources of randomness. Such tools are randomness extractors, procedures that convert biased distributions into almost uniform ones, and pseudorandom generators, procedures that stretch a short random input into a much longer output that has the property of being indistinguishable (a term that is given a precise technical meaning) from the uniformdistribution. The other theme of the research component is the study of probabilistically checkable proofs (PCP), a model of computation that gives a surprising characterization of NP in terms of efficient randomized proof-checking. The PCP model is the best known tool to prove results about the complexity of finding approximate solutions for NP-hard combinatorial optimization problems. The goal of this project is to look for stronger characterizations of NP in the PCP model, for more applications to the study of the approximability of optimization problems and, with special emphasis, for a simplified proof of the PCP characterization of NP, a result that currently has an exceedingly complicated proof.The educational component of this project will integrate material on randomized algorithms, pseudorandomness, and probabilistic proof-systems into existing courses on algorithms and complexity and into a new course on cryptography that the principal investigator is developing. A main goal of the educational component is to give elementary presentations of some results that have so far been confined to research-oriented graduate courses. This is unfortunate because they are relevant and entertaining, not particularly hard to explain, and can have a strong motivational influence. An extensive set of lecture notes will be developed on this material.

摘要：随机性的使用以戏剧性且尚未完全理解的方式影响计算：在算法设计中，它产生了更简单、更有效的方法来解决计算问题；在复杂性理论中，它提出了新的概念和模型，有时会带来意想不到的（且影响深远的）结果。该职业发展项目涉及一系列与计算随机性相关的研究和教育活动。该项目的研究部分涉及两个主题。目标之一是开发通用工具，使随机算法更加鲁棒，这样即使它们是使用有偏见的、有限的随机源来实现的，它们也能发挥作用。此类工具包括随机性提取器（将有偏差的分布转换为几乎均匀分布的过程）和伪随机生成器（将短随机输入拉伸为较长输出的过程），该输出具有与均匀分布无法区分的属性（该术语被赋予精确的技术含义）。研究部分的另一个主题是概率可检查证明（PCP）的研究，这是一种计算模型，在有效的随机证明检查方面给出了 NP 的令人惊讶的特征。 PCP 模型是最著名的工具，用于证明寻找 NP 难组合优化问题近似解的复杂性。该项目的目标是在 PCP 模型中寻找更强的 NP 表征，更多地应用于研究优化问题的近似性，并特别强调 NP 的 PCP 表征的简化证明，这一结果目前具有极其复杂的证明。该项目的教育部分将把有关随机算法、伪随机性和概率证明系统的材料整合到现有课程中 算法和复杂性以及首席研究员正在开发的密码学新课程。教育部分的一个主要目标是对迄今为止仅限于研究型研究生课程的一些成果进行基本介绍。 这是不幸的，因为它们是相关且有趣的，不是特别难以解释，并且可以产生强大的激励影响。 将根据该材料编写一套内容广泛的讲义。