发表机构
Lund University（隆德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种精确搜索 Moore 图的算法，通过多种工程优化将 k=9 的搜索时间从 47.6 秒降至 17.9 秒，并验证了 k=57 时内存占用较小。
AI 中文摘要
Moore 图同时要求两个通常相互矛盾的条件：每个顶点应有尽可能多的邻居，而任意两个顶点之间的距离应保持很小。对于直径为 2 且度为 $k$ 的情况，Moore 界为 $k^2+1$，达到该界将强制得到一个非常严格的图：它是一个强正则图，参数为 $(k^2+1,k,0,1)$。Hoffman 和 Singleton 证明了这样的图仅可能存在于 $k=2,3,7$ 以及可能的 57。度为 57 的情况涉及 3,250 个顶点，至今仍未解决。本文并非试图解决这个开放问题，而是提出一个更具算法性的问题：如何高效地精确搜索 Moore 结构本身？我们从基于半径二 Moore 树的人工开发的穷举搜索开始。叶子分为 $k$ 组，每组大小为 $k-1$，任意两组之间的边构成一个完美匹配——即一个置换。为了提高效率，当部分构造产生三角形或四边形时，立即放弃该构造。然后，我们描述了一系列通过人类引导的 AI 过程开发的精确算法工程改进，这些改进仅在数学验证和受控回归测试后被接受。主要步骤包括：可逆短环禁止计数器、位并行域、最小剩余值分支、单例传播、基于匹配的 AllDifferent 过滤、平衡可重放并行前沿、残差对称轨道剪枝以及前沿状态的规范去重。在主要的 $k=9$ 基准测试中，更强的对称性剪枝将六核笔记本电脑上的墙钟时间从 47.6 秒缩短至 17.9 秒。进一步的并行加速在另一份报告中给出；在 16 核/32 线程的机器上，中位墙钟时间降至 5.28 秒。度为 10 和 57 的情况用作压力测试，五分钟的度为 57 的运行仅使用 161 MiB 的峰值常驻内存，表明状态规模适中。
英文摘要
A Moore graph simultaneously asks for two things that normally pull in opposite directions: every vertex should have many neighbors, while every pair of vertices should remain close. For diameter two and degree $k$, the Moore bound is $k^2+1$, and equality forces a remarkably rigid graph: it is strongly regular with parameters $(k^2+1,k,0,1)$. Hoffman and Singleton showed that such a graph can exist only for $k=2,3,7$, and possibly 57. The degree-57 case on 3,250 vertices remains open. This paper is not an attempt to settle that open problem. Instead, we ask a more algorithmic question: How efficiently can the defining Moore structure itself be searched exactly? We begin from a human-developed exhaustive search based on the radius-two Moore tree. Leaves fall into $k$ groups of size $k-1$, and the edges between every pair of such groups form a perfect matching - a permutation. For efficiency, a partial construction is abandoned immediately when it creates a triangle or quadrangle. We then describe a sequence of exact algorithm-engineering improvements developed through a human-guided AI process and accepted only after mathematical checking and controlled regression tests. The main steps are reversible short-cycle forbid counters, bit-parallel domains, minimum-remaining-value branching, singleton propagation, matching-based AllDifferent filtering, balanced replayable parallel frontiers, residual-symmetry orbit pruning, and canonical deduplication of frontier states. On the principal $k=9$ benchmark, stronger symmetry cuts wall time from 47.6 to 17.9 seconds on a six-core laptop setup. Further parallelization speedup is reported separately; on a 16-core/32-thread machine, median wall time is reduced to 5.28 seconds. Degrees 10 and 57 are used as stress tests, and a five-minute degree-57 run uses only 161 MiB peak resident memory, showing that state size is modest.
