  We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and
$O(k^2)$. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
$\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.
