On the Complexity of Sub-Tree Scheduling for Wireless Sensor Networks with Partial Coverage

Given an undirected graph $G$ whose edge weights change over $s$ time slots, the sub-tree scheduling for wireless sensor networks with partial coverage asks to partition the vertices of $G$ in $s$ non-empty trees such that the total weight of the trees is minimized. In this note we show that the problem is NP-hard in both the cases where $s$ $(i)$ is part of the input and $(ii)$ is a fixed instance parameter. In both our proofs we reduce from the cardinality Steiner tree problem. We additionally give polynomial-time algorithms for structured inputs of the problem.