AF: Large: Collaborative Research: Exploiting Duality between Algorithms and Complexity

AF：大：协作研究：利用算法和复杂性之间的二元性

• 批准号: 1212372
• 负责人: Ryan Williams
• 金额: $ 45万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1212372&HistoricalAwards=false
• 关键词: AF; Large; Collaborative; Research; Exploiting

Meta-algorithms are algorithms that take other algorithms as input. Meta-algorithms are important in a variety of applications, from minimizing circuits in VLSI to verifying hardware and software to machine learning. Lower bound proofs show that computational problems are difficult in the sense of requiring a prohibitive amount of time, memory, or other resource to solve. This is particularly important in the context of cryptography, where it is vital to ensure that no feasible adversary can break a code. Surprisingly, recent research by the PIs and others shows that designing meta-algorithms is, in a formal sense, equivalent to proving lower bounds. In other words, one can prove a negative (the non-existence of a small circuit to solve a problem) by a positive (devising a new meta-algorithm). This was the key to a breakthrough by PI Williams, proving lower bounds on constant depth circuits with modular arithmetic gates. The proposed research will utilize this connection both to design new meta-algorithms and to prove new lower bounds. A primary focus will be on meta-algorithms for deciding if a given algorithm is 'trivial' or not, such as algorithms for the Boolean satisfiability problem. The proposed research will devise new algorithms that improve over exhaustive search for many variants of satisfiability. On the other hand, it will also explore complexity-theoretic limitations on how much improvement is possible, using reductions and lower bounds for restricted models. Satisfiability will provide a starting point for a more general understanding of the exact complexities of other NP-complete problems such as the traveling salesman problem and k-colorability. The proposal addresses both worst-case performance and the use of fast algorithms as heuristics for solving this problem. This exploration will be mainly mathematical. However, when new algorithms and heuristics are developed, they will be implemented and the resulting software made widely available. This research will be incorporated in courses taught by the PI's, at both graduate and undergraduate levels. Both graduate and undergraduate students will perform research as part of the project.

元算法是以其他算法作为输入的算法。元算法在各种应用中都很重要，从最小化 VLSI 中的电路到验证硬件和软件再到机器学习。下界证明表明，计算问题在需要大量时间、内存或其他资源来解决的意义上是困难的。这在密码学的背景下尤其重要，因为确保任何可行的对手都无法破解密码至关重要。令人惊讶的是，PI 和其他人最近的研究表明，设计元算法在形式上相当于证明下界。换句话说，人们可以通过肯定（设计一种新的元算法）来证明否定（不存在解决问题的小电路）。这是 PI Williams 突破的关键，证明了具有模块化算术门的恒定深度电路的下限。拟议的研究将利用这种联系来设计新的元算法并证明新的下限。主要关注点是用于确定给定算法是否“平凡”的元算法，例如布尔可满足性问题的算法。拟议的研究将设计新的算法，改进对可满足性的许多变体的过度搜索。另一方面，它还将探索复杂性理论的局限性，即使用受限模型的减少和下限来确定可能的改进程度。可满足性将为更全面地理解其他 NP 完全问题（例如旅行商问题和 k 着色性问题）的确切复杂性提供一个起点。该提案解决了最坏情况下的性能以及使用快速算法作为解决该问题的启发式方法。这种探索主要是数学方面的。然而，当新的算法和启发法被开发出来时，它们将被实施，并且由此产生的软件将被广泛使用。这项研究将纳入 PI 教授的研究生和本科生课程中。作为该项目的一部分，研究生和本科生都将进行研究。