Smallest Multiplicative Stretch for an $α$-APASP Below $\widetilde O \left(n^ω\right)$ Runtime

Dor, Halperin and Zwick [FOCS'96; SICOMP'00] showed that $(2-\varepsilon)$-APASP is at least as hard as BMM for every fixed $0<\varepsilon<1$. Zwick [FOCS'98; JACM'02] gave a $2$-APASP algorithm in $\widetilde O(n^{ω+o(1)}\log W_{\max})$ runtime, where $W_{\max}$ is the maximum edge weight. Bringmann, K{ü}nnemann and W{ę}grzycki [STOC'19] removed this logarithmic factor. Using OpenAI's recent bound $ω\leq\frac{9}{4}$ gives $\widetilde O\left(n^{\frac{9}{4}+o(1)}\right)$ time. We present two weighted undirected $2$-APASP algorithms. The first uses $(\min,\max)$-products. Earlier rectangular FMM bounds of Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'24] and Alman, Duan, Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'25] give $\widetilde O(n^{2.36999465})$ runtime, below the previous $ω\leq2.371177$ of Dupont, Eisenberger, Kozlovskii, Mehrabian, Ruiz, See, Zhou, Alman, Vassilevska Williams and Balog [arXiv'26]. Current rectangular FMM bounds give $\widetilde O(n^{2.34549151})$, independent of $W_{\max}$. For integer weights, our second algorithm has $\widetilde O(n^{ω-\frac{(ω-2)^2}{ω-1}+o(1)}\log^{O(1)}W_{\max})$ runtime, with exponent below $ω$ whenever $ω>2$. The rectangular bounds of Uzna{ń}ski [arXiv'26] or of Amanov, van den Berg and Zuiddam [arXiv'26] give $\widetilde O(n^{2.17157288}\log^{O(1)}W_{\max})$.