On k-column sparse packing programs

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 Ω(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 (e2 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. © 2010 Springer-Verlag.