  Relaxation and rounding approaches became a standard and extremely versatile
tool for constrained submodular function maximization. One of the most common
rounding techniques in this context are contention resolution schemes. Such
schemes round a fractional point by first rounding each coordinate
independently, and then dropping some elements to reach a feasible set. Also
the second step, where elements are dropped, is typically randomized. This
leads to an additional source of randomization within the procedure, which can
complicate the analysis. We suggest a different, polyhedral viewpoint to design
contention resolution schemes, which avoids to deal explicitly with the
randomization in the second step. This is achieved by focusing on the marginals
of a dropping procedure. Apart from avoiding one source of randomization, our
viewpoint allows for employing polyhedral techniques. Both can significantly
simplify the construction and analysis of contention resolution schemes. We
show how, through our framework, one can obtain an optimal monotone contention
resolution scheme for bipartite matchings. So far, only very few results are
known about optimality of monotone contention resolution schemes. Our
contention resolution scheme for the bipartite case also improves the lower
bound on the correlation gap for bipartite matchings. Furthermore, we derive a
monotone contention resolution scheme for matchings that significantly improves
over the previously best one. At the same time, our scheme implies that the
currently best lower bound on the correlation gap for matchings is not tight.
Our results lead to improved approximation factors for various constrained
submodular function maximization problems over a combination of matching
constraints with further constraints.
