最小扩张图扩展的双标准近似法

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Sampson Wong
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引用次数: 0

摘要

扳手构造侧重于网络的初始设计。然而,网络往往会随着时间的推移而不断改进。在本文中,我们将重点放在改进步骤上。给定一个图和 $k$ 预算,我们要在图中添加哪些 $k$ 边来最小化图的扩张?Gudmundsson 和 Wong [TALG'22]为这个问题提供了第一个正结果,但他们的近似系数与 $k$ 成线性关系。我们的主要结果是一个 $(2 \sqrt[r]{2} \ k^{1/r},2r)$ 双标准近似,对于所有 $r \geq 1$,运行时间为 $O(n^3 \log n)$。换句话说,如果$t^*$是在图中添加任意$k$边后的最小扩张,那么我们的算法在图中添加$O(k^{1+1/r})$边,就能得到$2rt^*$的扩张。此外,我们对算法的分析在 Erd\H{o}sgirth 猜想下是严密的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Bicriteria approximation for minimum dilation graph augmentation
Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget $k$, which $k$ edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in $k$. Our main result is a $(2 \sqrt[r]{2} \ k^{1/r},2r)$-bicriteria approximation that runs in $O(n^3 \log n)$ time, for all $r \geq 1$. In other words, if $t^*$ is the minimum dilation after adding any $k$ edges to a graph, then our algorithm adds $O(k^{1+1/r})$ edges to the graph to obtain a dilation of $2rt^*$. Moreover, our analysis of the algorithm is tight under the Erd\H{o}s girth conjecture.
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