使用遗产二分法近似生成树拥堵
Approximation of Spanning Tree Congestion using Hereditary Bisection
AI总结：
本文提出一种O(Δ·log^(3/2)n)的近似算法，改进了生成树拥堵问题的近似比，通过新的下界证明了生成树拥堵与遗产二分法的关系。
摘要：
生成树拥堵（STC）问题是NP难问题：给定图G，构造生成树T，使其最大边拥堵最小。边e∈T的拥堵是图G中满足T中u-v路径经过e的边uv的数量。已知每个生成树是n/2近似。本文提出O(Δ·log^(3/2)n)近似算法，对于最大度数为顶点数多项对数的图，优于现有最佳近似。主要工具是新的下界证明：对于图G，STC(G)≥Ω(hb(G)/Δ)，其中hb(G)是G的所有子图的最大二分宽。
英文摘要：
The Spanning Tree Congestion (STC) problem is the following NP-hard problem: given a graph $G$, construct a spanning tree $T$ of $G$ minimizing its maximum edge congestion where the congestion of an edge $e\in T$ is the number of edges $uv$ in $G$ such that the unique path between $u$ and $v$ in $T$ passes through $e$; the optimal value for a given graph $G$ is denoted $STC(G)$. It is known that every spanning tree is an $n/2$-approximation for the STP problem. A long-standing problem is to design a better approximation algorithm. Our contribution towards this goal is an $O(Δ\cdot\log^{3/2}n)$-approximation algorithm where $Δ$ is the maximum degree in $G$ and $n$ the number of vertices. For graphs with a maximum degree bounded by a polylog of the number of vertices, this is an exponential improvement over the previous best approximation. Our main tool for the algorithm is a new lower bound on the spanning tree congestion which is of independent interest. Denoting by $hb(G)$ the hereditary bisection of $G$ which is the maximum bisection width over all subgraphs of $G$, we prove that for every graph $G$, $STC(G)\geq Ω(hb(G)/Δ)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
