It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA’16 8 J.ACM’17, Bodwin-Williams SODA’16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O ( n 3/5 + ε )-approximation for distance preservers and pairwise spanners (for arbitrary constant ε > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O (log n )-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O (1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O ( n 3/5 + ε )-approximation for the Directed Steiner Forest problem (for arbitrary constant ε > 0) when all edges have uniform costs, improving the previous best O ( n 2/3 + ε )-approximation due to Berman et al. [ICALP’11] (which holds for general edge costs).

中文翻译：

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近似扳手和定向施泰纳森林

最近发现两者之间存在非常密切的联系加法扳手（所有距离都保留到附加拉伸的子图），距离保持器（需求对的距离精确保留的子图），和成对扳手（需求对的距离保持到乘法或加法拉伸的子图）[Abboud-Bodwin SODA'16 8 J.ACM'17，Bodwin-Williams SODA'16]。我们从优化的角度研究这些问题，而不是研究极值实例的存在，而是给我们一个实例，并要求我们找到最稀疏的 spanner/preserver。我们给出一个○(n 3/5 + ε) - 距离保持器和成对扳手的近似值（对于任意常数 ε > 0）。这是这两个问题的第一个非平凡上界，众所周知，这两个问题都与 Label Cover 一样难以近似。我们还证明了近似加法扳手的标签覆盖硬度，即使对于加法 1 拉伸的情况（人们可能期望多对数近似，因为相关的乘法 2 扳手问题承认○（日志n)-近似）和加法多对数拉伸（其中相关的乘法扳手问题具有○(1)-近似)。有趣的是，我们在近似算法中使用的技术从基于距离的问题扩展到纯粹的连接网络设计问题。特别是，我们的技术允许我们给出一个○(n 3/5 + ε)-当所有边具有统一成本时，有向施泰纳森林问题（对于任意常数 ε > 0）的近似值，改进了之前的最佳○(n 2/3 + ε) - 由 Berman 等人得出的近似值。[ICALP'11]（适用于一般边缘成本）。