On the Parameterized Complexity of Spanning Trees with Small Vertex Covers

We consider the minimum power spanning tree (MPST) problem with general and unit demands from a parameterized perspective. The case of unit demands is equivalent to the problem of finding a spanning tree with the smallest possible vertex cover (MCST). We show that MPST is W[1]-hard when parameterized by the vertex cover of the input graph, and is W[2]-hard when parameterized by the solution size—the latter holds even in the case of unit demands. For the special case of unit demands, however, we demonstrate an FPT algorithm when parameterized by treewidth. In the context of kernelization, we show that even MCST is unlikely to admit a polynomial kernel under standard complexity-theoretic assumptions when parameterized by the vertex cover of the input graph.