  The contention resolution framework is a versatile rounding technique used as
a part of the relaxation and rounding approach for solving constrained
submodular function maximization problems. We apply this framework to the
hypergraph matching, knapsack, and $k$-column sparse packing problems. In the
hypergraph matching setting, we adapt the technique of Guruganesh, Lee (2018)
to non-constructively prove that the correlation gap is at least
$\frac{1-e^{-k}}{k}$ and provide a monotone
$\left(b,\frac{1-e^{-bk}}{bk}\right)$-balanced contention resolution scheme,
generalizing the results of Bruggmann, Zenklusen (2019). For the knapsack
problem, we prove that the correlation gap of instances where exactly $k$
copies of each item fit into the knapsack is at least $\frac{1-e^{-2}}{2}$ and
provide several monotone contention resolution schemes: a
$\frac{1-e^{-2}}{2}$-balanced scheme for instances where all item sizes are
strictly bigger than $\frac{1}{2}$, a $\frac{4}{9}$-balanced scheme for
instances where all item sizes are at most $\frac{1}{2}$, and a
$0.279$-balanced scheme for instances with arbitrary item sizes. For $k$-column
sparse packing integer programs, we slightly modify the
$\left(2k+o\left(k\right)\right)$-approximation algorithm for $k$-CS-PIP based
on the strengthened LP relaxation presented in Brubach et al. (2019) to obtain
a $\frac{1}{4k+o\left(k\right)}$-balanced contention resolution scheme and
hence a $\left(4k+o\left(k\right)\right)$-approximation algorithm for
$k$-CS-PIP based on the natural LP relaxation.
