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

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.