  We consider approximation algorithms for packing integer programs (PIPs) of
the form $\max\{\langle c, x\rangle : Ax \le b, x \in \{0,1\}^n\}$ where $c$,
$A$, and $b$ are nonnegative. We let $W = \min_{i,j} b_i / A_{i,j}$ denote the
width of $A$ which is at least $1$. Previous work by Bansal et al.
\cite{bansal-sparse} obtained an $\Omega(\frac{1}{\Delta_0^{1/\lfloor W
\rfloor}})$-approximation ratio where $\Delta_0$ is the maximum number of
nonzeroes in any column of $A$ (in other words the $\ell_0$-column sparsity of
$A$). They raised the question of obtaining approximation ratios based on the
$\ell_1$-column sparsity of $A$ (denoted by $\Delta_1$) which can be much
smaller than $\Delta_0$. Motivated by recent work on covering integer programs
(CIPs) \cite{cq,chs-16} we show that simple algorithms based on randomized
rounding followed by alteration, similar to those of Bansal et al.
\cite{bansal-sparse} (but with a twist), yield approximation ratios for PIPs
based on $\Delta_1$. First, following an integrality gap example from
\cite{bansal-sparse}, we observe that the case of $W=1$ is as hard as maximum
independent set even when $\Delta_1 \le 2$. 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 $W = 1 + \epsilon$
where $\epsilon \in (0,1]$, we obtain an
$\Omega(\epsilon^2/\Delta_1)$-approximation. In the large width regime, when $W
\ge 2$, we obtain an $\Omega((\frac{1}{1 +
\Delta_1/W})^{1/(W-1)})$-approximation. We also obtain a
$(1-\epsilon)$-approximation when $W = \Omega(\frac{\log
(\Delta_1/\epsilon)}{\epsilon^2})$.
