Hop-Constrained Metric Embeddings and their Applications

TL;DR

This paper advances hop-constrained metric embeddings, improving their parameters and applying these results to develop better approximation algorithms and routing schemes for network design problems.

Contribution

It improves existing hop-constrained Ramsey-type embeddings, generalizes them for arbitrary distortion, and applies these to enhance algorithms for network design and routing.

Findings

Improved embedding parameters to t=β=O(log n)/ε.

Achieved polynomial improvements in approximation algorithms.

Constructed hop-constrained routing schemes with provable guarantees.

Abstract

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 $(t,M,\beta,h)$ (here $t,\beta,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $\beta$ 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 $1-\epsilon$ fraction of the vertices and $\beta=t=O(\frac{\log^2 n}{\epsilon})$. 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 $t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$, 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.