The "short cycle removal" technique was recently introduced by Abboud, Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of approximation. Its main technical result is that listing all triangles in an $n^{1/2}$-regular graph is $n^{2-o(1)}$-hard under the 3-SUM conjecture even when the number of short cycles is small; namely, when the number of $k$-cycles is $O(n^{k/2+\gamma})$ for $\gamma<1/2$. Abboud et al. achieve $\gamma\geq 1/4$ by applying structure vs. randomness arguments on graphs. In this paper, we take a step back and apply conceptually similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve the best possible $\gamma=0$ and the following lower bounds under the 3-SUM conjecture: * Approximate distance oracles: The seminal Thorup-Zwick distance oracles achieve stretch $2k\pm O(1)$ after preprocessing a graph in $O(m n^{1/k})$ time. For the same stretch, and assuming the query time is $n^{o(1)}$ Abboud et al. proved an $\Omega(m^{1+\frac{1}{12.7552 \cdot k}})$ lower bound on the preprocessing time; we improve it to $\Omega(m^{1+\frac1{2k}})$ which is only a factor 2 away from the upper bound. We also obtain tight bounds for stretch $2+o(1)$ and $3-\epsilon$ and higher lower bounds for dynamic shortest paths. * Listing 4-cycles: Abboud et al. proved the first super-linear lower bound for listing all 4-cycles in a graph, ruling out $(m^{1.1927}+t)^{1+o(1)}$ time algorithms where $t$ is the number of 4-cycles. We settle the complexity of this basic problem by showing that the $\widetilde{O}(\min(m^{4/3},n^2) +t)$ upper bound is tight up to $n^{o(1)}$ factors. Our results exploit a rich tool set from additive combinatorics, most notably the Balog-Szemer\'edi-Gowers theorem and Rusza's covering lemma. A key ingredient that may be of independent interest is a subquadratic algorithm for 3-SUM if one of the sets has small doubling.

翻译：“短环移除”技术由Abboud、Bringmann、Khoury和Zamir（STOC '22）近期提出，用于证明近似问题的细粒度困难性。其核心技术成果是：在3-SUM猜想下，即使短环数量较少，在$n^{1/2}$-正则图中列出所有三角形仍具有$n^{2-o(1)}$的难度——即当$k$-环数量为$O(n^{k/2+\gamma})$且$\gamma<1/2$时。Abboud等人通过对图应用结构与随机性论证，实现了$\gamma\geq 1/4$。本文退一步，对3-SUM问题的数值应用概念上类似的论证，从而实现了最优的$\gamma=0$，并在3-SUM猜想下得到以下下界： * 近似距离预言机：经典的Thorup-Zwick距离预言机在$O(m n^{1/k})$时间预处理图后，可实现拉伸$2k\pm O(1)$。对于相同拉伸且假设查询时间为$n^{o(1)}$，Abboud等人证明了预处理时间的$\Omega(m^{1+\frac{1}{12.7552 \cdot k}})$下界；我们将其改进为$\Omega(m^{1+\frac1{2k}})$，仅与上界相差2倍。我们还得到了拉伸$2+o(1)$和$3-\epsilon$的紧界，以及动态最短路径的更高下界。 * 列出4-环：Abboud等人首次证明了在图中列出所有4-环的次线性下界，排除了$(m^{1.1927}+t)^{1+o(1)}$时间算法（其中$t$是4-环数量）。我们通过证明$\widetilde{O}(\min(m^{4/3},n^2) +t)$上界在$n^{o(1)}$因子内是紧的，解决了这一基本问题的复杂度。 我们的结果利用了加性组合学中的丰富工具集，最著名的是Balog-Szemerédi-Gowers定理和Ruzsa覆盖引理。一个可能具有独立意义的关键要素是：当其中一个集合具有小加倍常数时，存在次二次时间的3-SUM算法。