We consider approximation algorithms for packing integer programs (PIPs) of the form where c, A, and b are nonnegative. We let denote the width of A which is at least 1. Previous work by Bansal et al.  obtained an -approximation ratio where is the maximum number of nonzeroes in any column of A (in other words the -column sparsity of A). They raised the question of obtaining approximation ratios based on the -column sparsity of A (denoted by) which can be much smaller than Motivated by recent work on covering integer programs (CIPs) [4, 7] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al.  (but with a twist), yield approximation ratios for PIPs based on First, following an integrality gap example from , we observe that the case of is as hard as maximum independent set even when In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width where we obtain an -approximation. In the large width regime, when we obtain an -approximation. We also obtain a -approximation when.