We consider approximation algorithms for packing integer programs (PIPs) of the form @math where @math , @math , and @math are nonnegative. We let @math denote the width of @math which is at least @math . Previous work by bansal-sparse obtained an @math -approximation ratio where @math is the maximum number of nonzeroes in any column of @math (in other words the @math -column sparsity of @math ). They raised the question of obtaining approximation ratios based on the @math -column sparsity of @math (denoted by @math ) which can be much smaller than @math . Motivated by recent work on covering integer programs (CIPs) cq,chs-16 we show that simple algorithms based on randomized rounding followed by alteration, similar to those of bansal-sparse (but with a twist), yield approximation ratios for PIPs based on @math . First, following an integrality gap example from bansal-sparse , we observe that the case of @math is as hard as maximum independent set even when @math . 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 @math where @math , we obtain an @math -approximation. In the large width regime, when @math , we obtain an @math -approximation. We also obtain a @math -approximation when @math .
A folklore result uses the Lovasz local lemma to analyze the discrepancy of hypergraphs with bounded degree and edge size. We generalize this result to the context of real matrices with bounded row and column sums. We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variables get assigned large (integral) values; Kolliopoulos & Young (2005) modified this algorithm to limit the solution size, at the cost of a worse approximation ratio. We develop a new rounding scheme based on the Partial Resampling variant of the Lovasz Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln([EQUATION]), where amin is the minimum covering constraint and Δ1 is the maximum e1-norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]); we also show nearly-matching inapproximability and integrality-gap lower bounds. Our approach improves asymptotically, in several different ways, over known results. First, it replaces Δ0, the maximum number of nonzeroes in any column (from the result of Srinivasan) by Δ1 which is always - and can be much - smaller than Δ0; this is the first such result in this context. Second, our algorithm automatically handles multi-criteria programs; we achieve improved approximation ratios compared to the algorithm of Srinivasan, and give, for the first time when the number of objective functions is large, polynomial-time algorithms with good multi-criteria approximations. We also significantly improve upon the upper-bounds of Kolliopoulos & Young when the integer variables are required to be within (1 + e) of some given upper-bounds, and show nearly-matching inapproximability.