We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless communication network: Given an edge-weighted $n$-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. On the negative side, we show that $o(\log n)$-approximating the difference $d$ between the optimal solution cost and a natural lower bound is NP-hard and that, under the Exponential Time Hypothesis, there are no exact algorithms running in $2^{o(n)}$ time or in $f(d)\cdot n^{O(1)}$ time for any computable function $f$. Moreover, we show that the special case of connecting $c$ network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size $c^{O(1)}$ unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects $O(\log n)$ connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components due to sensor faults. In experiments, the algorithm outperforms CPLEX with known ILP formulations when $n$ is sufficiently large compared to $c$.