  Until recently, LP relaxations have played a limited role in the design of
approximation algorithms for the Steiner tree problem. In 2010, Byrka et al.
presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation,
but surprisingly, their analysis does not provide a matching bound on the
integrality gap.
  We take a fresh look at hypergraphic LP relaxations for the Steiner tree
problem - one that heavily exploits methods and results from the theory of
matroids and submodular functions - which leads to stronger integrality gaps,
faster algorithms, and a variety of structural insights of independent
interest. More precisely, we present a deterministic ln(4)+epsilon
approximation that compares against the LP value and therefore proves a
matching ln(4) upper bound on the integrality gap.
  Similarly to Byrka et al., we iteratively fix one component and update the LP
solution. However, whereas they solve an LP at every iteration after
contracting a component, we show how feasibility can be maintained by a greedy
procedure on a well-chosen matroid. Apart from avoiding the expensive step of
solving a hypergraphic LP at each iteration, our algorithm can be analyzed
using a simple potential function. This gives an easy means to determine
stronger approximation guarantees and integrality gaps when considering
restricted graph topologies. In particular, this readily leads to a 73/60 bound
on the integrality gap for quasi-bipartite graphs.
  For the case of quasi-bipartite graphs, we present a simple algorithm to
transform an optimal solution to the bidirected cut relaxation to an optimal
solution of the hypergraphic relaxation, leading to a fast 73/60 approximation
for quasi-bipartite graphs. Furthermore, we show how the separation problem of
the hypergraphic relaxation can be solved by computing maximum flows, providing
a fast independence oracle for our matroids.
