Downgrading to Minimize Connectivity

We study the problem of interdicting a directed graph by deleting nodes with the goal of minimizing the local edge connectivity of the remaining graph from a given source to a sink. We show hardness of obtaining strictly unicriterion approximations for this basic vertex interdiction problem. We also introduce and study a general downgrading variant of the interdiction problem where the capacity of an arc is a function of the subset of its endpoints that are downgraded, and the goal is to minimize the downgraded capacity of a minimum source-sink cut subject to a node downgrading budget. This models the case when both ends of an arc must be downgraded to remove it, for example. For this generalization, we provide a bicriteria $(4,4)$-approximation that downgrades nodes with total weight at most 4 times the budget and provides a solution where the downgraded connectivity from the source to the sink is at most 4 times that in an optimal solution. WE accomplish this with an LP relaxation and round using a ball-growing algorithm based on the LP values. We further generalize the downgrading problem to one where each vertex can be downgraded to one of $k$ levels, and the arc capacities are functions of the pairs of levels to which its ends are downgraded. We generalize our LP rounding to get $(4k,4k)$-approximation for this case.