  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.
