We investigate structural properties of the binary rank of Kronecker powers of binary matrices, equivalently, the biclique partition numbers of the corresponding bipartite graphs. To this end, we engineer a Column Generation approach to solve linear optimization problems for the fractional biclique partition number of bipartite graphs, specifically examining the Domino graph and its Kronecker powers. We address the challenges posed by the double exponential growth of the number of bicliques in increasing Kronecker powers. We discuss various strategies to generate suitable initial sets of bicliques, including an inductive method for increasing Kronecker powers. We show how to manage the number of active bicliques to improve running time and to stay within memory limits. Our computational results reveal that the fractional binary rank is not multiplicative with respect to the Kronecker product. Hence, there are binary matrices, and bipartite graphs, respectively, such as the Domino, where the asymptotic fractional binary rank is strictly smaller than the fractional binary rank. While we used our algorithm to reduce the upper bound, we formally prove that the fractional biclique cover number is a lower bound, which is at least as good as the widely used isolating (or fooling set) bound. For the Domino, we obtain that the asymptotic fractional binary rank lies in the interval $[2,2.373]$. Since our computational resources are not sufficient to further reduce the upper bound, we encourage further exploration using more substantial computing resources or further mathematical engineering techniques to narrow the gap and advance our understanding of biclique partitions, particularly, to settle the open question whether binary rank and biclique partition number are multiplicative with respect to the Kronecker product.

翻译：本研究探讨了二进制矩阵克罗内克幂的二进制秩的结构性质，等价地，即对应二分图的双团划分数。为此，我们设计了一种列生成方法，用于求解二分图分数双团划分数（特别是多米诺图及其克罗内克幂）的线性优化问题。我们应对了在递增克罗内克幂中双团数量呈双指数增长所带来的挑战。我们讨论了生成合适初始双团集合的各种策略，包括针对递增克罗内克幂的归纳方法。我们展示了如何管理活跃双团的数量以改善运行时间并保持在内存限制内。我们的计算结果表明，分数二进制秩关于克罗内克积不满足乘法性。因此，存在二进制矩阵（及相应的二分图，例如多米诺图），其渐近分数二进制秩严格小于分数二进制秩。虽然我们使用算法降低了上界，但我们正式证明了分数双团覆盖数是一个下界，其效果至少与广泛使用的隔离集（或愚弄集）界相当。对于多米诺图，我们得出其渐近分数二进制秩位于区间 $[2,2.373]$ 内。由于我们的计算资源不足以进一步降低上界，我们鼓励利用更强大的计算资源或进一步的数学工程技术进行深入探索，以缩小该区间并增进对双团划分的理解，特别是为解决二进制秩和双团划分数关于克罗内克积是否满足乘法性这一开放性问题做出贡献。