π€ AI Summary
This study addresses the upper bounds on the complexity of rectangular matrix multiplication and the efficiency bottlenecks in all-pairs shortest path (APSP) algorithms. The proposed method extends the OpenAI framework to rectangular multiplication, establishing key bounds such as Ο(1,k,1)β€2. By integrating shared-leg entropy inequalities, polynomial degeneration, and tensor spectral duality techniques, it derives a double-exponential parameter Ξ±β₯1/2, which is subsequently leveraged to optimize Zwickβs algorithm. The primary contributions of this work include novel upper bounds for rectangular multiplication complexity and a theoretical breakthrough achieving an APSP running time of O(n^2.4999).
π Abstract
In this note, we extend the analysis underlying a recent matrix-multiplication result by OpenAI to rectangular products and prove that $Ο(1,k,1)\le 2$ for $0\le k\le \frac{1}{2}$ and $Ο(1,k,1)\le 1+k+\frac{1}{4k}$ for $k\ge \frac{1}{2}$. In particular, $Ο(1,\frac{1}{2},1)=2$ and the dual exponent satisfies $Ξ±\ge \frac{1}{2}$. We use the shared-leg entropy inequality and polynomial-multiplication degenerations from that work, retaining two-leg symmetry and the orientation of each sector. Logarithmic averaging produces homogeneous auxiliary profiles. Their powered versions have a common asymptotic slope, and bounding their intercepts gives the spectral constraint $b\le 4a(1-a)$. This yields the rectangular curve by tensor-spectrum duality. As an application, Zwick's algorithm for all-pairs shortest paths in directed unweighted graphs runs in $O(n^{2.5})$ time. Combining the rectangular bound with the $(\min,+)$-product improvement of Alman and Vassilevska Williams further gives $O(n^{2.4999})$ running time.