The all pairs shortest path problem (APSP) is one of the foundational problems in computer science. For weighted dense graphs on $n$ vertices, no truly sub-cubic algorithms exist to compute APSP exactly even for undirected graphs. This is popularly known as the APSP conjecture and has played a prominent role in developing the field of fine-grained complexity. The seminal result of Seidel uses fast matrix multiplication (FMM) to compute APSP on unweighted undirected graphs exactly in $\tilde{O}(n^{\omega})$ time, where $\omega=2.372$. Even for unweighted undirected graphs, it is not possible to obtain a $(2-\epsilon)$-approximation of APSP in $o(n^\omega)$ time. In this paper, we provide a multitude of new results for multiplicative and additive approximations of APSP in undirected graphs for both unweighted and weighted cases. We provide new algorithms for multiplicative 2-approximation of unweighted graphs: a deterministic one that runs in $\tilde{O}(n^{2.072})$ time and a randomized one that runs in $\tilde{O}(n^{2.032})$ on expectation improving upon the best known bound of $\tilde{O}(n^{2.25})$ by Roditty (STOC, 2023). For $2$-approximating paths of length $\geq k$, $k \geq 4$, we provide the first improvement after Dor, Halperin, Zwick (2000) for dense graphs even just using combinatorial methods, and then improve it further using FMM. We next consider additive approximations, and provide improved bounds for all additive $\beta$-approximations, $\beta \geq 4$. For weighted graphs, we show that by allowing small additive errors along with an $(1+\epsilon)$-multiplicative approximation, it is possible to improve upon Zwick's $\tilde{O}(n^\omega)$ algorithm. Our results point out the crucial role that FMM can play even on approximating APSP on unweighted undirected graphs, and reveal new bottlenecks towards achieving a quadratic running time to approximate APSP.

翻译：全对最短路径问题（APSP）是计算机科学中的基础问题之一。对于包含 $n$ 个顶点的加权稠密图，即使是无向图，目前也不存在真正次三次的精确求解APSP算法。这通常被称为APSP猜想，并在细粒度复杂性领域的发展中发挥了重要作用。Seidel的开创性工作利用快速矩阵乘法（FMM）在 $\tilde{O}(n^{\omega})$ 时间内精确求解了无权无向图的APSP，其中 $\omega=2.372$。即使对于无权无向图，也无法在 $o(n^\omega)$ 时间内获得 $(2-\epsilon)$ 近似的APSP。在本文中，我们针对无向图的APSP，在无权与加权两种情形下，提供了大量关于乘法与加法近似的新结果。对于无权图的乘法2-近似，我们提出了新算法：一个确定性算法运行时间为 $\tilde{O}(n^{2.072})$，以及一个期望运行时间为 $\tilde{O}(n^{2.032})$ 的随机算法，改进了Roditty（STOC，2023）已知最优的 $\tilde{O}(n^{2.25})$ 界。对于长度 $\geq k$（$k \geq 4$）路径的2-近似，我们首次在Dor、Halperin、Zwick（2000）之后实现了稠密图上的改进，即使仅使用组合方法，并进一步利用FMM加以提升。接下来我们考虑加法近似，并为所有加法 $\beta$-近似（$\beta \geq 4$）提供了改进的界。对于加权图，我们证明通过允许较小的加法误差以及 $(1+\epsilon)$ 乘法近似，可以改进Zwick的 $\tilde{O}(n^\omega)$ 算法。我们的结果指出了FMM在无权无向图APSP近似中可能发挥的关键作用，并揭示了实现APSP近似二次运行时间的新瓶颈。