  In the laminar-constrained spanning tree problem, the goal is to find a
minimum-cost spanning tree which respects upper bounds on the number of times
each cut in a given laminar family is crossed. This generalizes the
well-studied degree-bounded spanning tree problem, as well as a previously
studied setting where a chain of cuts is given. We give the first
constant-factor approximation algorithm; in particular we show how to obtain a
multiplicative violation of the crossing bounds of less than 22 while losing
less than a factor of 5 in terms of cost.
  Our result compares to the natural LP relaxation. As a consequence, our
results show that given a $k$-edge-connected graph and a laminar family
$\mathcal{L} \subseteq 2^V$ of cuts, there exists a spanning tree which
contains only an $O(1/k)$ fraction of the edges across every cut in
$\mathcal{L}$. This can be viewed as progress towards the Thin Tree Conjecture,
which (in a strong form) states that this guarantee can be obtained for all
cuts simultaneously.
