[论文解读] Methods for Solving Extremal Problems in Practice

Michael Codish|arXiv (Cornell University)|Aug 24, 2015

Coding theory and cryptography参考文献 4被引用 1

一句话总结

本文证明了某些7输入布尔函数的乘法复杂度至少为7，将M(7)的下界提高了1。通过基于电路拓扑和对称性约化的精细计数论证，作者表明6个与门不足以计算所有2^128种可能的7元布尔函数，从而证明在GF(2)上至少需要7次乘法才能实现某些函数。

ABSTRACT

During the 20 th century there has been an incredible progress in solving theoretically hard problems in practice. One of the most prominent examples is the DPLL algorithm and its derivatives to solve the Boolean satisfiability problem, which can handle instances with millions of variables and clauses in reasonable time, notwithstanding the theoretical difficulty of solving the problem. Despite this progress, there are classes of problems that contain especially hard instances, which have remained open for decades despite their relative small size. One such class is the class of extremal problems, which typically involve finding a combinatorial object under some constraints (e.g, the search for Ramsey numbers). In recent years, a number of specialized methods have emerged to tackle extremal problems. Most of these methods are applied to a specific problem, despite the fact there is a great deal in common between different problems. Following a meticulous examination of these methods, we would like to extend them to handle general extremal problems. Further more, we would like to offer ways to exploit the general structure of extremal problems in order to develop constraints and symmetry breaking techniques which will, hopefully, improve existing tools. The latter point is of immense importance in the context of extremal problems, which often hamper existing tools when there is a great deal of symmetry in the search space, or when not enough is known of the problem structure. For example, if a graph is a solution to a problem instance, in many cases any isomorphic graph will also be a solution. In such cases, existing methods can usually be applied only if the model excludes symmetries.

研究动机与目标

• 建立n元布尔函数最大乘法复杂度M(n)的更紧下界。
• 解决M(7) > 6这一长期悬而未决的开放问题，尽管此前已有相关界限。
• 开发一种系统方法，利用电路拓扑对固定数量与门下可计算函数的数量进行计数。
• 通过对称性破缺和拓扑剪枝减少等价电路的搜索空间，实现高效枚举。
• 提供一种非构造性证明，证明存在需要7个与门的7输入布尔函数，未来工作将致力于构造此类函数。

提出的方法

• 将电路抽象为仅捕捉与门之间互连关系的拓扑结构，忽略异或门的细节。
• 引入良层化和最小拓扑的概念，以减少电路结构中的冗余和等价性。
• 应用生成与剪枝算法，枚举k个与门下非等价的最小良层化拓扑。
• 使用定理4将可计算布尔函数的数量上界表示为3^k × 2^{2kn + n + k + 1} × |Tk/≡|，其中|Tk/≡|为非等价拓扑的数量。
• 应用鸽巢原理：由于7元布尔函数的数量（2^128）超过使用6个与门可计算函数的上界，因此至少存在一个函数需要7个与门。
• 采用对称性破缺技术，确保拓扑的规范表示，减少重复计数，并实现|Tk/≡|的高效计算。

实验结果

研究问题

• RQ1 7输入布尔函数的乘法复杂度是否至少为7？
• RQ2 能否将使用6个与门可计算的布尔函数数量紧致地界定，以证明并非所有7元函数都可计算？
• RQ3 对于具有k个与门的电路，非等价最小良层化拓扑的数量是多少？该数量如何减少搜索空间？
• RQ4 能否通过非构造性证明确立存在一个乘法复杂度为7的7输入函数？
• RQ5 M(6)的值是多少？能否通过精细化计算方法确定该值？

主要发现

• 本文证明了M(7) ≥ 7，将此前已知的下界提高了1。
• 通过生成算法计算得出，具有6个与门的电路共有555,709个非等价最小良层化拓扑。
• 使用6个与门可计算的7元布尔函数数量的上界小于2^128，具体为小于2^128，证明并非所有此类函数都可计算。
• 该上界推导为555,709 × 3^6 × 2^98 < 2^128，表明6个与门不足以计算所有7输入函数。
• 该结果为非构造性：它证明了存在一个乘法复杂度为7的7输入布尔函数，但并未显式构造出该函数。
• 该方法表明，当n=7时，M(n) > n−1成立，这是目前所知该严格不等式成立的最小情况。