改进的欧几里得浅光树
Improved Euclidean Shallow Light Trees
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中文总结 AI 辅助
该研究解决了欧几里得平面中浅光树的长期问题，构造出根伸展为1+ε、亮度远优于前人结果的浅光树，突破了2/ε的亮度壁垒，接近已知下界。
中文摘要 AI 辅助
对于参数α,β≥1，以指定顶点r为根的带权图G的生成树T称为(α,β)-浅光树（SLT），当满足：(i)对每个顶点v，d_T(r,v)≤α·d_G(r,v)（根伸展为α）；(ii)w(T)≤β·w(MST)（亮度为β）。Khuller、Raghavachari和Young的开创性工作（SODA 1993）为一般带权图构造了(1+ε,2/ε+1)-SLT，并证明该根伸展与亮度的权衡关系即使对串并联图也是紧的。他们进一步提出问题：在欧几里得平面中，是否能实现亮度的轻微改进，即对任意常数c>0将亮度降至(2−c)/ε？我们肯定地解决了这个长期存在的问题。具体而言，我们证明每个欧几里得实例都存在根伸展为1+ε、亮度至多为(5/3 + o_ε(1))·1/ε的SLT，显著突破了长期存在的2/ε壁垒。作为第二个主要结果，我们提供欧几里得平面中SLT的构造，其根伸展为1+ε、亮度至多为(2π/√(4π²+1)+o_ε(1))·1/ε≈(0.987+o_ε(1))·1/ε。值得注意的是，这将亮度界中的主导项2/ε减少了一倍以上，且非常接近Elkin和Solomon（FOCS 2011）给出的下界(2π/(2π+1)+o_ε(1))·1/ε≈(0.862+o_ε(1))·1/ε。
英文摘要
For parameters $α,β\geq 1$, a spanning tree $T$ of a weighted graph $G$ rooted at a designated vertex $r$ is called an $(α,β)$-shallow-light tree (SLT) if (i) for every vertex $v$, $d_T(r,v) \leq α\cdot d_G(r,v)$ (root-stretch $α$), and (ii) $w(T) \leq β\cdot w(\mathsf{MST})$ (lightness $β$). The pioneering work of Khuller, Raghavachari, and Young (SODA 1993) constructed $\left(1+ε, \tfrac{2}ε+1\right)$-SLTs for general weighted graphs, and proved that this tradeoff between root-stretch and lightness is tight even for series-parallel graphs. They further asked whether even a slight improvement, namely reducing the lightness to $\tfrac{2-c}ε$ for any constant $c>0$, is possible in the Euclidean plane. We resolve this longstanding question in the affirmative. Specifically, we show that every Euclidean instance admits an SLT with root-stretch $1+ε$ and lightness at most $\left(\frac{5}{3} + o_ε(1)\right) \cdot \frac{1}ε$, thereby significantly improving upon the longstanding $2/ε$ barrier. As our second main result, we provide a construction of SLTs in the Euclidean plane, with root stretch $1+ε$ and lightness at most $\left(\frac{2π}{\sqrt{4π^2+1}}+o_ε(1)\right)\frac{1}ε \approx (0.987+o_ε(1))\frac{1}ε$. Notably, this reduces the leading $2/ε$ term in the lightness bound by more than a factor of two, and comes quite close to the lower bound of $\left(\frac{2π}{2π+1} +o_ε(1))\right) \cdot \frac{1}ε \approx (0.862 +o_ε(1))\frac{1}ε$ by Elkin and Solomon (FOCS 2011).
