APSP with small integer weights in undirected graphs [Seidel'95, Galil and Margalit'97] has an $\tilde{O}(n^\omega)$ time algorithm, where $\omega<2.373$ is the matrix multiplication exponent. APSP in directed graphs with small weights however, has a much slower running time that would be $\Omega(n^{2.5})$ even if $\omega=2$ [Zwick'02]. To understand this $n^{2.5}$ bottleneck, we build a web of reductions around directed unweighted APSP. We show that it is fine-grained equivalent to computing a rectangular Min-Plus product for matrices with integer entries; the dimensions and entry size of the matrices depend on the value of $\omega$. As a consequence, we establish an equivalence between APSP in directed unweighted graphs, APSP in directed graphs with small $(\tilde{O}(1))$ integer weights, All-Pairs Longest Paths in DAGs with small weights, approximate APSP with additive error $c$ in directed graphs with small weights, for $c\le \tilde{O}(1)$ and several other graph problems. We also provide fine-grained reductions from directed unweighted APSP to All-Pairs Shortest Lightest Paths (APSLP) in undirected graphs with $\{0,1\}$ weights and $\#_{\text{mod}\ c}$APSP in directed unweighted graphs (computing counts mod $c$). We complement our hardness results with new algorithms. We improve the known algorithms for APSLP in directed graphs with small integer weights and for approximate APSP with sublinear additive error in directed unweighted graphs. Our algorithm for approximate APSP with sublinear additive error is optimal, when viewed as a reduction to Min-Plus product. We also give new algorithms for variants of #APSP in unweighted graphs, as well as a near-optimal $\tilde{O}(n^3)$-time algorithm for the original #APSP problem in unweighted graphs. Our techniques also lead to a simpler alternative for the original APSP problem in undirected graphs with small integer weights.

中文翻译：

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全对最短路径的小权重变体的算法，归约和等价

在无向图中[Seidel'95，Galil和Margalit'97]具有小的整数权重的APSP具有$ \ tilde {O}（n ^ \ omega）$时间算法，其中$ \ omega <2.373 $是矩阵乘法指数。但是，有向图中具有较小权重的APSP的运行时间要慢得多，即使$ \ omega = 2 $ [Zwick'02]，运行时间也将是$ \ Omega（n ^ {2.5}）$。为了理解这个$ n ^ {2.5} $的瓶颈，我们围绕有针对性的未加权APSP建立了一个减少量的网。我们表明，它等效于为具有整数项的矩阵计算矩形Min-Plus乘积；矩阵的维数和入口大小取决于$ \ omega $的值。因此，我们在有向未加权图中的APSP，在具有较小$（\ tilde {O}（1））$整数权重的有向图中的APSP之间建立了等价关系，DAG中所有权重的最长路径，权重较小的有向图中的近似APSP，加性误差为$ c $，适用于$ c \ le \ tilde {O}（1）$和其他一些图形问题。我们还提供了权重为$ \ {0,1 \} $和$ \ #_ {\ text {mod} \ c} $的无向图中从有向无权APSP到全对最短最短路径（APSLP）的细化缩减。有向无加权图中的APSP（计算计数为$ c $模）。我们用新算法来补充硬度结果。我们改进了针对具有小整数权重的有向图中APSLP的已知算法，以及有向无权图中有亚线性累加误差的近似APSP的已知算法。当被视为Min-Plus乘积的减少时，我们的具有亚线性加法误差的近似APSP算法是最佳的。我们还为未加权图中的#APSP变体提供了新算法，并为未加权图中的原始#APSP问题提供了接近最优的$ \ tilde {O}（n ^ 3）$-time算法。我们的技术还为整数权重小的无向图中的原始APSP问题提供了一种更简单的替代方法。