发表机构
Ben-Gurion University（本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出 Steiner $k$-Forest 问题的 $n^{0.3+\varepsilon}$ 近似算法，改进已有结果，并基于 Densest $k$-Subgraph 下界证明 $O(\sqrt{k})$ 近似难以改进。
AI 中文摘要
我们针对 Steiner $k$-Forest 问题给出一个 $n^{0.3+\varepsilon}$ 近似算法，其中 $\varepsilon>0$ 为任意常数。作为 $n$ 的函数，这改进了 Gupta 等人 [ESA'07, TALG'10] 的 $O(\min\{\sqrt{n},\sqrt{k}\})$ 近似，该结果在一般情形下已保持近二十年，同时也改进了 Dinitz 等人 [APPROX-RANDOM'14, TALG'17] 针对均匀权重情形的 $n^{0.448}$ 近似。另一方面，我们证明，由于 Densest $k$-Subgraph 问题的下界，Steiner $k$-Forest 的 $O(\sqrt{k})$ 近似很可能无法改进。具体而言，我们证明对于任意足够小的 $\varepsilon>0$，一个 $O(k^{1/2-\varepsilon})$ 近似的 Steiner $k$-Forest 算法将超越已知的 Densest k-Subgraph 的度数 $n^{\Omega(\varepsilon^2)}$ Sum-of-Squares 完整性间隙，并反驳相应的稠密与随机猜想。
英文摘要
We give an $n^{0.3+\varepsilon}$-approximation algorithm for the Steiner $k$-Forest problem, for any constant $\varepsilon>0$. As a function of $n$, this improves over the $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation of Gupta et al. [ESA'07, TALG'10] which has stood for nearly two decades for the general case, as well as the later $n^{0.448}$-approximation of Dinitz et. al [APPROX-RANDOM'14, TALG'17] for the uniform weight case. On the other hand, we show that, due to lower bounds on the Densest $k$-Subgraph problem, the $O(\sqrt k)$-approximation for Steiner $k$-Forest likely cannot be improved. Specifically, we show that for any sufficiently small $\varepsilon>0$, an $O(k^{1/2-\varepsilon})$-approximation for Steiner $k$-Forest would surpass known degree-$n^{Ω(\varepsilon^2)}$ Sum-of-Squares integrality gaps for Densest k-Subgraph and refute the corresponding dense-versus-random conjecture.
