A classic 1993 paper by Alth\H{o}fer et al. proved a tight reduction from spanners, emulators, and distance oracles to the extremal function $\gamma$ of high-girth graphs. This paper initiated a large body of work in network design, in which problems are attacked by reduction to $\gamma$ or the analogous extremal function for other girth concepts. In this paper, we introduce and study a new girth concept that we call the bridge girth of path systems, and we show that it can be used to significantly expand and improve this web of connections between girth problems and network design. We prove two kinds of results: 1) We write the maximum possible size of an $n$-node, $p$-path system with bridge girth $>k$ as $\beta(n, p, k)$, and we write a certain variant for "ordered" path systems as $\beta^*(n, p, k)$. We identify several arguments in the literature that implicitly show upper or lower bounds on $\beta, \beta^*$, and we provide some polynomially improvements to these bounds. In particular, we construct a tight lower bound for $\beta(n, p, 2)$, and we polynomially improve the upper bounds for $\beta(n, p, 4)$ and $\beta^*(n, p, \infty)$. 2) We show that many state-of-the-art results in network design can be recovered or improved via black-box reductions to $\beta$ or $\beta^*$. Examples include bounds for distance/reachability preservers, exact hopsets, shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an integrality gap for directed Steiner forest. We believe that the concept of bridge girth can lead to a stronger and more organized map of the research area. Towards this, we leave many open problems, related to both bridge girth reductions and extremal bounds on the size of path systems with high bridge girth.

翻译：Althöfer等人1993年的经典论文证明了从spanners、emulators和distance oracles到高围图极值函数γ的紧致归约。该论文开创了网络设计领域的大量研究，其中通过归约到γ或其他围概念的类似极值函数来解决相关问题。本文引入并研究了一种全新的围概念——路径系统的桥围(bridge girth)，并证明其能够显著扩展和改善围问题与网络设计之间的关联网络。我们证明了两类结果：1) 定义具有桥围>k的n节点p路径系统的最大可能规模为β(n, p, k)，并定义其"有序"路径系统的变体为β*(n, p, k)。我们识别了文献中隐式给出β、β*上下界的若干论证，并对这些界提供了多项式改进。特别地，我们为β(n, p, 2)构造了紧致下界，并对β(n, p, 4)和β*(n, p, ∞)的上界进行了多项式改进。2) 我们证明许多网络设计中的前沿结果可通过黑盒归约到β或β*得到复现或改进。示例包括距离/可达性保持器、精确跳集、捷径集、有向多割与最稀疏割的流割间隙、以及有向Steiner森林的整数性间隙。我们相信桥围概念能够为研究领域提供更强且组织更清晰的图谱。为此，我们留下许多开放问题，涉及桥围归约与高桥围路径系统规模的极值界。