Unit Perturbations in Budgeted Spanning Tree Problems

The minimum spanning tree of a graph is a well-studied structure that is the basis of countless graph theoretic and optimization problem. We study the minimum spanning tree (MST) perturbation problem where the goal is to spend a fixed budget to increase the weight of edges in order to increase the weight of the MST as much as possible. Two popular models of perturbation are bulk and continuous. In the bulk model, the weight of any edge can be increased exactly once to some predetermined weight. In the continuous model, one can pay a fractional amount of cost to increase the weight of any edge by a proportional amount. Frederickson and Solis-Oba \cite{FS96} have studied these two models and showed that bulk perturbation for MST is as hard as the $k$-cut problem while the continuous perturbation model is solvable in poly-time. In this paper, we study an intermediate unit perturbation variation of this problem where the weight of each edge can be increased many times but at an integral unit amount every time. We provide an $(opt/2 -1)$-approximation in polynomial time where $opt$ is the optimal increase in the weight. We also study the associated dual targeted version of the problem where the goal is to increase the weight of the MST by a target amount while minimizing the cost of perturbation. We provide a $2$-approximation for this variation. Furthermore we show that assuming the Small Set Expansion Hypothesis, both problems are hard to approximate. We also point out an error in the proof provided by Frederickson and Solis-Oba in \cite{FS96} with regard to their solution to the continuous perturbation model. Although their algorithm is correct, their analysis is flawed. We provide a correct proof here.