Minimum Cost Globally Rigid Subgraphs

Abstract

A d-dimensional framework is a pair (G, p), where $G=(V,E)$ is a graph and p is a map from V to $\mathbb {R}^d$. The length of an edge of G is equal to the distance between the points corresponding to its end-vertices. The framework is said to be globally rigid if its edge lengths uniquely determine all pairwise distances in the framework. A graph G is called globally rigid in $\mathbb {R}^d$ if every generic d-dimensional framework (G, p) is globally rigid. Global rigidity has applications in wireless sensor network localization, molecular conformation, formation control, CAD, and elsewhere. Motivated by these applications we consider the following optimization problem: given a graph $G=(V,E)$, a non-negative cost function $c:E\rightarrow \mathbb {R}_{+}$ on the edge set of G, and a positive integer d. Find a subgraph $H=(V,E')$ of G, on the same vertex set, which is globally rigid in $\mathbb {R}^d$ and for which the total cost $c(E'):=\sum _{e\in E'} c(e)$ of the edges is as small as possible. This problem is NP-hard for all $d\ge 1$, even if c is uniform or G is complete and c is metric. We focus on the two-dimensional case, where we give $\frac{3}{2}$-approximation (resp. 2-approximation) algorithms for the uniform cost and metric versions. We also develop a constant factor approximation algorithm for the metric version of the d-dimensional problem, for every $d\ge 3$.