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 formula -hard to approximate.
We generalize our result to the case of maximizing monotone submodular functions over k-column sparse packing constraints, and obtain an formula-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.
|Title of host publication||Integer Programming and Combinatorial Optimization (14th International Conference, IPCO 2010, Lausanne, Switzerland, June 9-11, 2010. Proceedings)|
|Editors||F. Eisenbrand, F.B. Shepherd|
|Place of Publication||Berlin|
|Publication status||Published - 2010|
|Name||Lecture Notes in Computer Science|