In this paper we carefully combine Fredman's trick [SICOMP'76] and Matou\v{s}ek's approach for dominance product [IPL'91] to obtain powerful results in fine-grained complexity: - Under the hypothesis that APSP for undirected graphs with edge weights in $\{1, 2, \ldots, n\}$ requires $n^{3-o(1)}$ time (when $\omega=2$), we show a variety of conditional lower bounds, including an $n^{7/3-o(1)}$ lower bound for unweighted directed APSP and an $n^{2.2-o(1)}$ lower bound for computing the Minimum Witness Product between two $n \times n$ Boolean matrices, even if $\omega=2$, improving upon their trivial $n^2$ lower bounds. Our techniques can also be used to reduce the unweighted directed APSP problem to other problems. In particular, we show that (when $\omega = 2$), if unweighted directed APSP requires $n^{2.5-o(1)}$ time, then Minimum Witness Product requires $n^{7/3-o(1)}$ time. - We show that, surprisingly, many central problems in fine-grained complexity are equivalent to their natural counting versions. In particular, we show that Min-Plus Product and Exact Triangle are subcubically equivalent to their counting versions, and 3SUM is subquadratically equivalent to its counting version. - We obtain new algorithms using new variants of the Balog-Szemer\'edi-Gowers theorem from additive combinatorics. For example, we get an $O(n^{3.83})$ time deterministic algorithm for exactly counting the number of shortest paths in an arbitrary weighted graph, improving the textbook $\widetilde{O}(n^{4})$ time algorithm. We also get faster algorithms for 3SUM in preprocessed universes, and deterministic algorithms for 3SUM on monotone sets in $\{1, 2, \ldots, n\}^d$.

翻译：本文精细地结合了弗雷德曼技巧[SICOMP'76]与Matoušek关于支配乘积的方法[IPL'91]，在高细粒度复杂度领域取得了强有力的成果：- 在假设边权属于{1,2,…,n}的无向图APSP问题需要n^{3-o(1)}时间（当ω=2时）的条件下，我们证明了多种条件性下界，包括无权重有向APSP问题的n^{7/3-o(1)}下界，以及计算两个n×n布尔矩阵间最小见证乘积的n^{2.2-o(1)}下界（即使ω=2），这改进了其平凡的n^2下界。我们的技术还可用于将无权重有向APSP问题归约至其他问题。特别地，我们证明（当ω=2时），若无权重有向APSP需要n^{2.5-o(1)}时间，则最小见证乘积需要n^{7/3-o(1)}时间。- 我们出人意料地证明，高细粒度复杂度中的许多核心问题与其自然计数版本等价。具体而言，我们证明Min-Plus乘积和精确三角形问题与它们的计数版本是次立方等价的，而3SUM与其计数版本是次二次等价的。- 我们利用加法组合学中Balog-Szemerédi-Gowers定理的新变体获得了新算法。例如，我们提出了一种O(n^{3.83})时间的确定性算法，用于精确计算任意带权图中最短路径的数量，改进了教科书中的Õ(n^4)时间算法。我们还获得了预处理论域中3SUM问题的更快算法，以及{1,2,…,n}^d上单调集3SUM问题的确定性算法。