Globally linked pairs and cheapest globally rigid supergraphs

Given a graph $G$, a cost function on the non-edges of $G$, and an integer $d$, the problem of finding a cheapest globally rigid supergraph of $G$ in $\mathbb{R}^d$ is NP-hard for $d\geq 1$. For this problem, which is a common generalization of several well-studied graph augmentation problems, no approximation algorithm has previously been known for $d\geq 2$. Our main algorithmic result is a 5-approximation algorithm in the $d=2$ case. We achieve this by proving numerous new structural results on rigid graphs and globally linked vertex pairs. In particular, we show that every rigid graph in $\mathbb{R}^2$ has a tree-like structure, which conveys all the information regarding its globally rigid augmentations. Our results also yield a new, simple solution to the minimum cardinality version (where the cost function is uniform) for rigid input graphs, a problem which is known to be solvable in polynomial time.