最优非自适应视点选择
Optimal Non-Adaptive Vantage Point Selection
- Rutgers University（罗格斯大学）
- University of Michigan（密歇根大学）
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究视点选择问题，提出非自适应算法并证明最优竞争比为$\tilde{\Theta}(n^{2/3})$，并推广到多查询设置，给出匹配的复杂界限。
AI中文摘要：
我们研究视点选择问题，该问题由Ashvinkumar、Chowdhury、Gao、Goswami、Mitchell和Polishchuk在[WADS'25]中提出，用于建模互联网上瓶颈容量估计问题。输入是一个具有唯一最短路径的加权无向图，其中每条边都有一个不同的未知容量。当算法查询一个顶点$v$时，它会揭示从$v$到从$v$可达的每个其他顶点的最短路径上的最小容量边。目标是最大化被揭示的边的总数。算法的质量通过其与一个事先知道所有边容量的最优算法的竞争比来衡量。我们首先考虑基础的单查询设置，其中算法和最优算法都被限制为一次查询。竞争比有一个平凡的$O(n)$上界，而已知的最佳下界是$\tilde{\Omega}(\sqrt{n})$。我们提供了一个算法和匹配的下界（在多项式对数因子内），表明最佳可能的竞争比是$\tilde{\Theta}(n^{2/3})$。此外，我们将结果扩展到一般设置，其中最优算法允许$k$次查询，而我们的算法允许$\alpha k$次查询，其中$\alpha\geq 1$。我们提出了一个随机非自适应算法和匹配的下界（在多项式对数因子内），表明非自适应算法的最佳可能期望竞争比是以下令人惊讶的复杂界限：$$ \tilde{\Theta}\left( \min\left\{ \frac{n}{\alpha k}, \max\left( \sqrt{\frac{n}{\alpha}}, \frac{n^{2/3}}{\alpha k^{1/3}} \right) \right\} \right). $$
英文摘要：
We study the \emph{vantage point selection} problem, introduced by Ashvinkumar, Chowdhury, Gao, Goswami, Mitchell, and Polishchuk [WADS'25] to model the problem of estimating bottleneck capacities on the Internet. The input is a weighted undirected graph with unique shortest paths where every edge has a distinct unknown \emph{capacity}. When the algorithm \emph{queries} a vertex $v$, it reveals the minimum-capacity edge on the shortest path from $v$ to every other vertex reachable from $v$. The goal is to maximize the total number of revealed edges. The quality of an algorithm is measured by its competitive ratio against an optimal algorithm that knows all edge capacities a priori. We first consider the foundational single-query setting, where both the algorithm and the optimal algorithm are restricted to a single query. There is a trivial upper bound of $O(n)$ on the competitive ratio and the best known lower bound was $\tildeΩ(\sqrt{n})$. We provide an algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible competitive ratio is $\tildeΘ(n^{2/3})$. Furthermore, we extend our results to the general setting where the optimal algorithm is allowed $k$ queries and our algorithm is allowed $αk$ queries for $α\geq 1$. We present a randomized non-adaptive algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible expected competitive ratio for non-adaptive algorithms is the following surprisingly complex bound: $$ \tildeΘ\left( \min\left\{ \frac{n}{αk}, \max\left( \sqrt{\frac{n}α}, \frac{n^{2/3}}{αk^{1/3}} \right) \right\} \right). $$
