AI 中文总结
本文提出了一种在近线性时间内实现(1+ε,1)-近似的最小度数生成树算法，改进了现有方法的性能界。
AI 中文摘要
最小度生成树问题是经典的NP难问题，自20世纪90年代初以来其最优近似保证已确立。Fürer和Raghavachari给出了一个$\tilde O(mn)$时间算法，计算最大度为$\Delta^\star+1$的生成树。三十多年来，能否在近线性时间实现类似强保证一直未解决。我们给出首个近线性时间算法，在$\tilde O(m/\epsilon^2)$时间内计算最大度为$\lceil (1+\epsilon)\Delta^\star\rceil+1$的生成树。此前近线性时间算法要么界弱，要么要求稠密图。用同一框架，还能在$\tilde O(mn^{2/3})$时间计算最大度为$\Delta^\star+1$出的生成树，改进了现有算法。
英文摘要
The minimum degree spanning tree problem is a classic NP-hard problem whose optimal approximation guarantee was established since the early 1990s: Fürer and Raghavachari [FR92] gave an $\tilde O(mn)$-time algorithm that computes a spanning tree with maximum degree $Δ^\star+1$, where $Δ^\star$ denotes the optimum value. Whether similarly strong guarantees can be achieved in near-linear time has remained open for over three decades. We give the first near-linear-time algorithm that computes a spanning tree with maximum degree $\lceil (1+ε)Δ^\star\rceil+1$ in $\tilde O(m/ε^2)$ time. Prior near-linear-time algorithms either achieved the weaker bound $\lceil (1+ε)Δ^\star\rceil + O(\log n/ε^2)$ [DHZ20] or required dense graphs with $m\ge n^{7/4}$ [CQT21,BFW26]. Using the same framework, our algorithm can also compute a spanning tree with maximum degree $Δ^\star+1$ in $\tilde O(mn^{2/3})$ time, improving upon the recent $\tilde O(mn^{3/4})$-time algorithm of [BFW26]. These two results strictly improve all previous construction algorithms for the minimum degree spanning tree problem.
