  In this paper we use polarization to study the behavior of the depth and
regularity of a monomial ideal $I$, locally at a variable $x_i$, when we lower
the degree of all the highest powers of the variable $x_i$ occurring in the
minimal generating set of $I$, and examine the depth and regularity of powers
of edge ideals of clutters using combinatorial optimization techniques. If $I$
is the edge ideal of an unmixed clutter with the max-flow min-cut property, we
show that the powers of $I$ have non-increasing depth and non-decreasing
regularity. In particular edge ideals of unmixed bipartite graphs have
non-decreasing regularity. We are able to show that the symbolic powers of the
ideal of covers of the clique clutter of a strongly perfect graph have
non-increasing depth. A similar result holds for the ideal of covers of a
uniform ideal clutter.

Commutative Algebra;Combinatorics