The cage problem concerns finding $(k,g)$-graphs, which are $k$-regular graphs with girth $g$, of the smallest possible number of vertices. The central goal is to determine $n(k,g)$, the minimum order of such a graph, and to identify corresponding extremal graphs. In this paper, we study the cage problem and several of its variants from a computational perspective. Four complementary graph generation algorithms are developed based on exhaustive generation of lifts, a tabu search heuristic, a hill climbing heuristic and excision techniques. Using these methods, we establish new upper bounds for eleven cases of the classical cage problem: $n(3,16) \leq 936$, $n(3,17) \leq 2048$, $n(4,9) \leq 270$, $n(4,10) \leq 320$, $n(4,11) \leq 713$, $n(5,9) \leq 1116$, $n(6,11) \leq 7783$, $n(8,7) \leq 774$, $n(10,7) \leq 1608$, $n(12,7) \leq 2890$ and $n(14,7) \leq 4716$. Notably, our results improve upon several of the best-known bounds, some of which have stood unchanged for 22 years. Moreover, the improvement for $n(4,10)$, from the longstanding upper bound of 384 down to 320, is surprising and constitutes a substantial improvement. While the main focus is on the cage problem, we also adapted our algorithms for variants of the cage problem that received attention in the literature. For these variants, additional improvements are obtained, further narrowing the gaps between known lower and upper bounds.

翻译：笼图问题关注寻找具有最小可能顶点数的$(k,g)$-图，即度为$k$、围长为$g$的正则图。核心目标是确定此类图的最小阶数$n(k,g)$，并识别相应的极值图。本文从计算角度研究笼图问题及其若干变体。基于升图的穷举生成、禁忌搜索启发式、爬山启发式及切除技术，开发了四种互补的图生成算法。利用这些方法，我们为经典笼图问题的十一个情形确立了新的上界：$n(3,16) \\leq 936$，$n(3,17) \\leq 2048$，$n(4,9) \\leq 270$，$n(4,10) \\leq 320$，$n(4,11) \\leq 713$，$n(5,9) \\leq 1116$，$n(6,11) \\leq 7783$，$n(8,7) \\leq 774$，$n(10,7) \\leq 1608$，$n(12,7) \\leq 2890$及$n(14,7) \\leq 4716$。值得注意的是，我们的结果改进了多个已知最佳上界，其中部分上界已保持22年未变。此外，$n(4,10)$的上界从长期保持的384降至320，这一改进令人意外且幅度显著。虽然研究重点在于笼图问题，我们也调整了算法以处理文献中受关注的笼图问题变体。针对这些变体，我们获得了进一步的改进，进一步缩小了已知下界与上界之间的差距。