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：本文提出"自适应翻转图算法"，将自适应搜索与翻转图算法相结合，用于寻找快速高效的矩阵乘法方法。该算法针对原始翻转图算法在处理大规模矩阵乘法时存在的探索局限性及搜索效率低下问题进行了改进。在探索局限性方面，所提算法通过自适应地在翻转图上进行状态迁移，引入了一种不会严格减少乘法次数的灵活性。针对大规模实例中搜索效率低下的问题，该算法自适应地约束搜索范围，而非依赖完全随机的搜索方式，从而促进更有效的探索。数值实验结果表明了自适应翻转图算法的有效性：将$4\times 5$矩阵与$5\times 5$矩阵相乘的乘法次数从76次降至73次，并将$5 \times 5$矩阵与另一个$5\times 5$矩阵相乘的乘法次数从95次降至94次。以上结果均在特征为二的域中获得。