This study proposes the "adaptive flip graph algorithm", which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the inherent limitations of exploration and inefficient search encountered in the original flip graph algorithm, particularly when dealing with large matrix multiplication. For the limitation of exploration, the proposed algorithm adaptively transitions over the flip graph, introducing a flexibility that does not strictly reduce the number of multiplications. Concerning the issue of inefficient search in large instances, the proposed algorithm adaptively constraints the search range instead of relying on a completely random search, facilitating more effective exploration. Numerical experimental results demonstrate the effectiveness of the adaptive flip graph algorithm, showing a reduction in the number of multiplications for a $4\times 5$ matrix multiplied by a $5\times 5$ matrix from $76$ to $73$, and that from $95$ to $94$ for a $5 \times 5$ matrix multiplied by another $5\times 5$ matrix. These results are obtained in characteristic two.

翻译：本研究提出“自适应翻转图算法”，该算法将自适应搜索与翻转图算法相结合，以寻找快速高效的矩阵乘法方法。自适应翻转图算法解决了原始翻转图算法在探索深度和搜索效率方面的固有局限性，尤其针对大规模矩阵乘法的场景。针对探索深度不足的问题，所提算法在翻转图上进行自适应转移，引入灵活性，从而不严格限制乘法次数。为应对大规模实例中的低效搜索问题，该算法自适应地约束搜索范围，而非依赖完全随机搜索，从而促进更有效的探索。数值实验结果表明，自适应翻转图算法在处理$4\times5$矩阵与$5\times5$矩阵乘法时，乘法次数从$76$减少至$73$；在处理$5\times5$矩阵与$5\times5$矩阵乘法时，乘法次数从$95$减少至$94$。上述结果在特征数为二的域中实现。