  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.
