Approximating Spanning Trees with Low Crossing Number

We present a linear programming based algorithm for computing a spanning tree $T$ of a set $P$ of $n$ points in $\Re^d$, such that its crossing number is $O(\min(t \log n, n^{1-1/d}))$, where $t$ the minimum crossing number of any spanning tree of $P$. This is the first guaranteed approximation algorithm for this problem. We provide a similar approximation algorithm for the more general settings of building a spanning tree for a set system with bounded \VC dimension. Our approach is an alternative to the reweighting technique previously used in computing such spanning trees. Our approach is an alternative to the reweighting technique previously used in computing such spanning trees.