平面图中长度受限的网络设计
Length-Constrained Network Design in Planar Digraphs
浏览论文内容
中文总结 AI 辅助
研究平面图中长度受限的定向斯坦纳树和定向斯坦纳森林问题，给出多对数双标准近似算法，其近似比率与平面图中对应问题最佳已知比率匹配，长度约束有O(log k)违反，还得到批量购买问题的多对数近似。
中文摘要 AI 辅助
我们研究平面图中定向斯坦纳树（DST）和定向斯坦纳森林（DSF）的长度受限推广。在这两个问题中，输入是带边成本的有向图。DST要求找到连接根节点到给定终端集的最小成本子图，DSF要求找到连接给定源 - 汇终端对中每对的最小成本子图。在长度受限设置下，每条边都有成本和长度，输入包括长度界限h，目标是找到通过长度至多为h的路径连接每个终端对的最小成本子图。我们的工作受近期结果启发，给出了平面图中长度受限的DST和DSF的多对数双标准近似算法，其近似比率与平面图中DST和DSF的最佳已知比率匹配，长度约束有O(log k)的违反，其中k表示终端数（或终端对数）。作为推论，我们得到了平面图中批量购买DST和DSF的多对数近似。
英文摘要
We study length-constrained generalizations of Directed Steiner Tree (DST) and Directed Steiner Forest (DSF) in planar digraphs. In both problems, the input is a directed graph with edge costs. DST asks for a min-cost subgraph connecting a root to a given set of terminals, and DSF asks for a min-cost subgraph connecting each of a given set of source-sink terminal pairs. In the length-constrained setting, each edge has both a cost and a length, and the input includes a length bound $h$; the goal is to find a min-cost subgraph connecting each terminal pair via a path of length at most $h$. Our work is motivated by a recent line of results showing that several network design problems that are traditionally hard in directed graphs admit polylogarithmic approximation ratios in planar digraphs. We give polylogarithmic bicriteria approximation algorithms for length-constrained analogues of DST and DSF in planar digraphs. Our approximation ratios match the best known for DST and DSF in planar digraphs, with an $O(\log k)$ violation of the length constraint, where $k$ denotes the number of terminals (or terminal pairs). As corollaries, we obtain polylogarithmic approximations for buy-at-bulk DST and DSF in planar digraphs.
