We give new approximation algorithms for packing integer programs (PIPs) by employing the method of randomized rounding combined with alterations. Our first result is a simpler approximation algorithm for general PIPs which matches the best known bounds, and which admits an efficient parallel implementation. We also extend these results to a multi-criteria version of PIPs. Our second result is for the class of packing integer programs (PIPs) that are column sparse, i. e., where 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 over previously known O(k2)-approximation ratios. We also generalize our result to the case of maximizing non-negative monotone submodular functions over k-column sparse packing constraints, and obtain an ( e2k e−1 + o(k))-approximation algorithm. In obtaining this result, we prove a new property of submodular functions that generalizes the fractional subadditivity property, which might be of independent interest
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