We investigate the problem of path selection in radio networks for a given static set of $n$ sites in two- and three-dimensional space. For static point-to-point communication we define measures for congestion, dilation, and energy consumption that take interferences among communication links into account.
We show that energy-optimal path selection for radio networks can be computed in polynomial time. Then we introduce the diversity $g(V)$ of a set $V\subseteq \REAL^{\ddim}$ for any constant $\ddim$. It can be used to upper bound the number of interfering edges. For real-world applications it can be regarded as $\Theta(\log n)$. A main result is that a $c$-spanner construction as a communication network allows one to approximate the congestion-optimal path system by a factor of $O(g(V)^2)$.
Furthermore, we show that there are vertex sets where only one of the performance parameters congestion, dilation, and energy can be optimized at a time. We show trade-offs lower bounding congestion $\times$ dilation and dilation $\times$ energy. The trade-off between congestion and dilation increases with switching from two-dimensional to three-dimensional space. For congestion and energy the situation is even worse. It is only possible to find a reasonable approximation for either congestion or energy minimization, while the other parameter is at least a polynomial factor worse than in the optimal network.