Approximation Algorithms for Hop Constrained and Buy-at-Bulk Network Design via Hop Constrained Oblivious Routing

TL;DR

This paper develops new approximation algorithms for hop-constrained and buy-at-bulk network design problems, improving bounds and addressing fault-tolerance, using LP relaxations and recent routing techniques.

Contribution

It introduces a polylogarithmic approximation via LP relaxation for buy-at-bulk network design and extends fault-tolerant algorithms to multicommodity settings.

Findings

New polylogarithmic approximation for buy-at-bulk network design.

LP-based bicriteria algorithms for fault-tolerant hop-constrained networks.

Addresses open questions on integrality gaps and fault-tolerance in network design.

Abstract

We consider two-cost network design models in which edges of the input graph have an associated cost and length. We build upon recent advances in hop-constrained oblivious routing to obtain two sets of results. We address multicommodity buy-at-bulk network design in the nonuniform setting. Existing poly-logarithmic approximations are based on the junction tree approach [CHKS09,KN11]. We obtain a new polylogarithmic approximation via a natural LP relaxation. This establishes an upper bound on its integrality gap and affirmatively answers an open question raised in [CHKS09]. The rounding is based on recent results in hop-constrained oblivious routing [GHZ21], and this technique yields a polylogarithmic approximation in more general settings such as set connectivity. Our algorithm for buy-at-bulk network design is based on an LP-based reduction to hop constrained network design for which we obtain LP-based bicriteria approximation algorithms. We also consider a fault-tolerant version of hop constrained network design where one wants to design a low-cost network to guarantee short paths between a given set of source-sink pairs even when k-1 edges can fail. This model has been considered in network design [GL17,GML18,AJL20] but no approximation algorithms were known. We obtain polylogarithmic bicriteria approximation algorithms for the single-source setting for any fixed k. We build upon the single-source algorithm and the junction-tree approach to obtain an approximation algorithm for the multicommodity setting when at most one edge can fail.