We consider the class of packing integer programs (PIPs) that are column sparse, where there is a specified upper bound k on the number of constraints that each variable appears in. We give an improved (ek + o(k))-approximation algorithm for k-column sparse PIPs. Our algorithm is based on a linear programming relaxation, and involves randomized rounding combined with alteration. We also show that the integrality gap of our LP relaxation is at least 2k − 1; it is known that even special cases of k-column sparse PIPs are Ω( k log k )-hard to approximate. We generalize our result to the case of maximizing monotone submodular functions over kcolumn sparse packing constraints, and obtain an e 2 k e−1 + o(k) -approximation algorithm. In obtaining this result, we prove a new property of submodular functions that generalizes the fractionally subadditive property, which might be of independent interest. When the capacities of all constraints are large relative to the sizes, we obtain substantially better guarantees for these k-column sparse packing problems; again our result is tight (up to constant factors) relative to the natural LP relaxation.
Bansal et al. REF devised a deterministic 8k-approximation and a randomized (ek + o(k))-approximation for the more general problem of approximating k-column-sparse packing integer programs.
On k-Column Sparse Packing Programs
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Computer Science Mathematics