AI 中文总结
研究利用对称字典序对称子集反向搜索框架枚举有限集对称可行子集，针对余电路、电路和三角剖分应用提出新方法，经C++实现后大幅提速，还计算出多个新数量并发现相关实例问题。
AI 中文摘要
本文介绍、分析并应用了枚举框架对称字典序对称子集反向搜索的变体，用于枚举有限集的对称可行子集直至对称。该框架针对余电路、电路和整数配置的三角剖分这三个应用进行了详细实现。提出并分析了两种检查子集中字典序最小性的新方法：关键元素法和改进的切换表法。此外，还引入了新的依赖于应用的方法来减少必要的枚举节点数量：余电路的秩修剪和三角剖分的字典序修剪。通过在软件包TOPCOM中用C++实现这些想法，在所有三个应用中，已知基准测试都能大幅更快地计算出来。首次计算出了以下新数字（等等）：9维立方体的余电路数量、8维立方体的电路数量、5维与3维单纯形乘积的所有三角剖分数量，以及具有17个点且翻转图不连通的6维点配置的所有三角剖分数量（由Santos构造）。此外，对于Santos的三角剖分，通过计算检查发现其翻转图组件确实是纯非正则的。此外，在另一个5维有26个点的实例（也由Santos构造）中，发现了一个缺陷：Santos的三角剖分可以通过启发式方法翻转到原始点配置中的正则三角剖分。在点配置的轻微修改版本中，如果启发式方法不能再将Santos的三角剖分翻转到正则三角剖分。
英文摘要
This paper introduces, analyzes, and applies variants of the enumeration framework symmetric lexicographic symmetric-subset reverse search for the enumeration of symmetric feasible subsets of a finite set up to symmetry. The framework is implemented in detail for three applications: cocircuits, circuits, and triangulations of point configurations. There are two new methods presented and analyzed to check the lexicographic minimality of a subset in its orbit: the critical-element method and the modified switch-table method. Moreover, new application-dependent methods to reduce the number of necessary enumeration nodes are introduced: rank-pruning for cocircuits and lex-pruning for triangulations. With a C++-implementation of the ideas in the software package TOPCOM, in all three applications known benchmarks can be computed faster by a large margin. The following new numbers could be computed for the first time (among others): the number of cocircuits of the 9-cube, the number of circuits of the 8-cube, and the number of all triangulations of the product of a 5- and a 3-simplex, as well as the number of all triangulations of a point configuration in dimension six with 17~points with disconnected flip-graph (constructed by Santos). Moreover, for Santos's triangulation it has computationally been checked that its flip-graph component is indeed purely non-regular. Furthermore, in another instance in dimension five with 26 points (also constructed by Santos), a flaw has been detected: Santos's triangulation can be heuristically flipped to a regular triangulation in the original point configuration. In a mildly modified version of the point configuration, the heuristics cannot flip Santos's triangulation to a regular triangulation anymore.
Comments101 pages, submitted; fixed a typo in the abstract; embedded the licensing info in the document
