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

AF：大：协作研究：利用元算法和复杂性之间的二元性

• 批准号: 1213151
• 负责人: Russell Impagliazzo
• 金额: $ 125万
• 依托单位: University of California-San Diego
• 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 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1213151&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, fromminimizing circuits in VLSI to verifying hardware and software tomachine learning. Lower bound proofs show that computational problemsare difficult in the sense of requiring a prohibitiveamount 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 breaka code. Surprisingly, recent research by the PIs and othersshows 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 todesign new meta-algorithms and to prove new lower bounds.A primary focus will be on meta-algorithms fordeciding if a given algorithm is 'trivial' or not, such as algorithmsfor the Boolean satisfiability problem. The proposed research will devise newalgorithms that improve over exhaustive search for many variantsof satisfiability. On the other hand, it will also explorecomplexity-theoretic limitations on how much improvement ispossible, using reductions and lower bounds for restrictedmodels. Satisfiability will provide a starting point for a moregeneral understanding of the exact complexities of other NP-completeproblems such as the traveling salesman problem and k-colorability.The proposal addresses both worst-case performance and the useof fast algorithms as heuristics for solving this problem.This exploration will be mainly mathematical. However, whennew algorithms and heuristics are developed, they will beimplemented and the resulting software made widely available.This research will be incorporated in courses taught bythe PI's, at both graduate and undergraduate levels.Both graduate and undergraduate students will perform researchas 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，在密码学中，确保没有可行的对手能够破解代码至关重要。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，用模块化算术门证明了恒定深度电路的下界。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界。主要关注点将是用于确定给定算法是否“平凡”的元算法，例如布尔可满足性的算法 问题。 拟议的研究将设计新的算法来改进对许多可满足性变体的详尽搜索。 另一方面，它还将探索复杂性理论的局限性，即使用限制模型的约简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-可色性）的确切复杂性提供一个起点。该提案解决了最坏情况的性能以及使用快速算法作为解决该问题的启发式方法。这种探索将主要是数学方面的。 然而，当新的算法和启发式方法被开发出来时，它们将被实施，并且由此产生的软件将被广泛使用。这项研究将被纳入 PI 教授的研究生和本科生课程中。研究生和本科生都将作为该项目的一部分进行研究。