Abstract
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 \(\Omega(\frac{k}{\log k})\)-hard to approximate.
We generalize our result to the case of maximizing monotone submodular functions over k-column sparse packing constraints, and obtain an \(\smash{\left(\frac{e^2k}{e-1} + o(k) \right)}\)-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.
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Bansal, N., Korula, N., Nagarajan, V., Srinivasan, A. (2010). On k-Column Sparse Packing Programs. In: Eisenbrand, F., Shepherd, F.B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2010. Lecture Notes in Computer Science, vol 6080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13036-6_28
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DOI: https://doi.org/10.1007/978-3-642-13036-6_28
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