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.

翻译：APSP 在未定向的图形[Seidel'95, Galil 和 Margalit'97] 中有小整数重量的 APSP 。 要理解这个 $@tilde{O}(n ⁇ omega) 时间算法, 美元=2. 373美元是矩阵倍增。 然而, 在带有小重量的定向图形中, APSP 运行时间要慢得多, 美元=2美元 [Zwick'02] 。 为了理解这个 $_n%q%2.5} blockneck, 我们在未加权的 APSP 时间算法里, 美元=$2xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx