跳约束度量嵌入及其应用
Hop-Constrained Metric Embeddings and their Applications
AI总结：
本文改进跳约束Ramsey型树嵌入，参数达到t=β=Õ(logn)/ε，并推广到任意失真t，从而多项式改进近似算法；进一步构造跳约束氏族嵌入，用于群斯坦纳树问题的双准则近似算法；最后构建跳约束距离预言、距离标签及首个有保证的跳约束紧凑路由方案。
摘要：
在网络设计问题（如紧凑路由）中，目标是通过（近似的）最短路径在节点间路由数据包。这些路由的一个理想属性是跳数少，这使它们更可靠并降低传输成本。继随机树嵌入在算法设计中的巨大成功之后，Haeupler、Hershkowitz和Zuzic（STOC'21）研究了跳约束Ramsey型度量嵌入到树中。具体来说，嵌入$f:G(V,E)\rightarrow T$具有Ramsey跳失真$(t,M,\beta,h)$（这里$t,\beta,h\ge1$且$M\subseteq V$），如果$\forall u,v\in M$，$d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$。$t$称为失真，$\beta$称为跳拉伸，$d_G^{(h)}(u,v)$表示最多$h$跳的$u-v$路径的最小权重。Haeupler等人构造了嵌入，其中$M$包含$1-\epsilon$比例的顶点，且$\beta=t=O(\frac{\log^2 n}{\epsilon})$。他们利用该嵌入获得了跳约束网络设计问题的多个双准则近似算法。在本文中，我们首先改进Ramsey型嵌入，得到参数$t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$，并将其推广到任意失真参数$t$（以减小$M$大小为代价）。该嵌入立即对Haeupler等人的所有近似算法实现了多项式改进。此外，我们构造了跳约束氏族嵌入（其中每个顶点有多个副本），并利用它们构造群斯坦纳树问题的双准则近似算法，匹配了无约束版本的最新成果。最后，我们利用嵌入结果构造了跳约束距离预言、距离标签，以及最重要的——首个具有可证明保证的跳约束紧凑路由方案。
英文摘要：
In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,β,h)$ (here $t,β,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(β\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $β$ is called the hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ path with at most $h$ hops. Haeupler {\em et al.} constructed embedding where $M$ contains $1-ε$ fraction of the vertices and $β=t=O(\frac{\log^2 n}ε)$. They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems. In this paper, we first improve the Ramsey-type embedding to obtain parameters $t=β=\frac{\tilde{O}(\log n)}ε$, and generalize it to arbitrary distortion parameter $t$ (in the cost of reducing the size of $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler {\em et al.}. Further, we construct hop-constrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees.
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