Approximating Spanning Trees with Low Crossing Number

TL;DR

This paper introduces the first guaranteed approximation algorithm for constructing spanning trees with low crossing numbers in high-dimensional point sets, using linear programming instead of reweighting techniques.

Contribution

It presents a novel LP-based approximation algorithm for spanning trees with low crossing numbers, applicable to sets with bounded VC dimension, offering an alternative to existing reweighting methods.

Findings

Achieves $O( ext{min}(t \log n, n^{1-1/d}))$ crossing number bound

First guaranteed approximation algorithm for this problem

Extends to set systems with bounded VC dimension

Abstract

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.