SIAM Journal on Computing, Volume 52, Issue 3, Page 718-739, June 2023.
Abstract. The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem ([math]) is arguably one of the most basic problems in this area: given a [math](-edge)-connected graph [math] and a set of extra edges (links), select a minimum cardinality subset [math] of links such that adding [math] to [math] increases its edge connectivity to [math]. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and Jájá, SIAM J. Comput., 10 (1981), pp. 270–283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290–306] that [math] can be reduced to the case [math], also known as the tree augmentation problem ([math]) for odd [math], and to the case [math], also known as the cactus augmentation problem ([math]) for even [math]. Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC’20, ACM, New York, 2020, pp. 815–825], several better than 2 approximation algorithms were known for [math], culminating with a recent [math] approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC’18, ACM, New York, 1918, pp. 632–645]. However, for [math] the best known approximation was 2. In this paper we breach the 2 approximation barrier for [math], hence, for [math], by presenting a polynomial-time [math] approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for [math] either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP ’14, Springer, Berlin, 2014, pp. 800–811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we “open the box” and exploit the specific properties of the instances of a Steiner tree arising from [math]. In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area.

中文翻译：

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突破连接增强的 2 近似障碍：斯坦纳树的简化

SIAM 计算杂志，第 52 卷，第 3 期，第 718-739 页，2023 年 6 月。
抽象的。可生存网络设计的基本目标是构建一个廉价的网络，尽管少数边缘/节点发生故障，但仍能维持给定节点集之间的连接。连接增强问题（[math]）可以说是该领域最基本的问题之一：给定一个[math]（-edge）连接图[math]和一组额外的边（链接），选择一个最小基数链接的子集 [math]，这样将 [math] 添加到 [math] 会增加其与 [math] 的边缘连接性。直觉上，人们希望通过增加额外的边缘来增强现有网络的可靠性。这个 NP 困难问题最著名的近似因子是 2，这可以通过多种方法来实现（第一个这样的结果在 [GN Frederickson and Jájá, SIAM J. Comput., 10 (1981), pp. 270– 283].众所周知[EA Dinitz, AV Karzanov, 和 MV Lomonosov，《离散优化研究》，Nauka，莫斯科，1976 年，第 290–306 页] 可以将 [math] 简化为情况 [math]，也称为奇数的树增广问题 ([math]) math]，对于 [math] 的情况，也称为偶数 [math] 的仙人掌增强问题 ([math])。在本文的会议版本之前 [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC'20, ACM, New York, 2020, pp. 815–825]，几种比 2 近似算法更好的[数学]，最终以最近的[数学]近似[F. Grandoni、C. Kalaitzis 和 R. Zenklusen，STOC'18，ACM，纽约，1918 年，第 632–645 页]。然而，对于[数学]来说，最著名的近似值是2。在本文中，我们突破了[数学]的2近似障碍，因此，对于[数学]，通过提出多项式时间[数学]近似值。从技术角度来看，我们的方法与以前的工作有很大不同。特别是，[数学] 的优于 2 的近似算法要么利用贪婪式算法，要么基于精心设计的 LP 舍入。相反，我们使用了先前在参数化算法中使用的 Steiner 树问题的简化方法 [Basavaraju 等人，ICALP '14，Springer，Berlin，2014，第 800–811 页]。这种减少不是近似保留，并且使用 Steiner 树的当前最佳近似因子 [Byrka et al., J. ACM, 60 (2013), 6] 作为黑盒不足以改进 2。为了实现后一个目标，我们“打开盒子”并利用[数学]产生的斯坦纳树实例的特定属性。