最小斯坦纳点树的精确算法
Exact Algorithms for Minimum Steiner Point Trees
AI总结：
针对平面$L_p$度量下带最小斯坦纳点数量与有界边长约束的斯坦纳树问题，提出一种确定性精确算法，可在$n^{O(n)}$时间内求解最优解并给出值界。
AI中文摘要：
给定平面上互不相同的终端点集$P\boldsymbol{\u2282}\boldsymbol{R}^2$和正数$R$，具有最小斯坦纳点数量与有界边长的斯坦纳树问题要求构造一棵连接$P$的直线树，每条边长度不超过$R$，并最小化斯坦纳点的数量。边长按固定的$L_p$度量计算，其中$p$属于非负有理数集或无穷大。最优斯坦纳点数量$k$不受$n$（终端点数量）限制，即使在仅两个终端点的情况下亦是如此。本文提出一种确定性精确算法，在计算模型下以$n^{O(n)}$的时间复杂度计算最优隐式表示，该表示与$k$无关。此表示包含完整斯坦纳拓扑结构、精确分支坐标及每条拓扑边的线段计数，细分操作需额外$\boldsymbol{\u0398}(n+k)$时间。对于每个完整斯坦纳拓扑，可行的线段计数向量是$O(n)$维凸投影的整数点，连续松弛将整数最优解限制为$2n-3$个连续值，通过精确半代数例程与积分格坐标下的平坦度递归来确定这些值。结合Bandyapadhyay等人的参数化瓶颈算法，可为本文考虑的每个固定度量提供值界$\boldsymbol{\u006d}\boldsymbol{\u0069}\boldsymbol{\u006e}\boldsymbol{\u007b}\boldsymbol{\u006e}^{O(n)}\boldsymbol{,}\boldsymbol{\u006b}^{O(k)}\boldsymbol{n}^{O(1)}\boldsymbol{\u007d}$。
英文摘要：
Given distinct terminals $P\subset R^2$ and $R>0$, the Steiner tree problem with minimum number of Steiner points and bounded edge length asks for a straight line tree spanning $P$, with every edge of length at most $R$, that minimizes the number of Steiner points. Length is measured in a fixed $L_p$ metric with $p\in Q_{\ge 1}\cup\{\infty\}$. The optimum $k$ is not bounded by $n$, even in two-terminal case. We give a deterministic exact algorithm that computes an optimal implicit representation in $n^{O(n)}$ time, independent of $k$, in the computation model of Section~\ref{subseccomputation}. The representation consists of a full Steiner topology, exact branch coordinates, and a segment count for each topology edge. Subdivision requires additional time $Θ(n+k)$. For each full Steiner topology, the feasible segment count vectors are the integer points of a convex projection in $O(n)$ dimensions. A continuous relaxation restricts the integer optimum to $2n-3$ consecutive values. Exact semialgebraic routines and a flatness recursion in integral lattice coordinates decide these values. Together with the parameterized bottleneck algorithm of Bandyapadhyay et al., this gives the value bound $\min\{n^{O(n)}, k^{O(k)}n^{O(1)}\}$ for every fixed metric considered here.
