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, making them more reliable and reducing transmission costs. Following the success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees.
Specifically, an embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,\beta,h)$ (where $t,\beta,h\ge1$ and $M\subseteq V$) if for all $u,v\in M$, $d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. Here, $t$ is called the distortion, $\beta$ is the hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ path with at most $h$ hops.
Haeupler et al. constructed an embedding where $M$ contains a $1-\epsilon$ fraction of the vertices and $\beta=t=O(\frac{\log^2 n}{\epsilon})$. They used their embedding to derive multiple bicriteria approximation algorithms for hop-constrained network design problems.
In this paper, we first improve the Ramsey-type embedding to obtain parameters $t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$ and generalize it to arbitrary distortion parameter $t$ (at the cost of reducing the size of $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler et al.
Furthermore, we construct hop-constrained clan embeddings (where each vertex has multiple copies) and use them to develop bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the unconstrained version.
Finally, we leverage our embedding results to construct hop-constrained distance oracles, distance labeling, and notably, the first hop-constrained compact routing scheme with provable guarantees.
Blogger's Review: This paper demonstrates significant advancements in network design by improving hop-constrained metric embeddings. The introduction of new parameters not only optimizes existing algorithms but also offers fresh perspectives for addressing complex network issues, particularly enhancing routing efficiency and reliability in practical applications.