On k-Column Sparse Packing Programs

TitleOn k-Column Sparse Packing Programs
Publication TypeBook Chapters
Year of Publication2010
AuthorsBansal N, Korula N, Nagarajan V, Srinivasan A
EditorEisenbrand F, Shepherd F
Book TitleInteger Programming and Combinatorial OptimizationInteger Programming and Combinatorial Optimization
Series TitleLecture Notes in Computer Science
Pagination369 - 382
PublisherSpringer Berlin / Heidelberg
ISBN Number978-3-642-13035-9

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 (klogk)-hard to approximate.We generalize our result to the case of maximizing monotone submodular functions over k-column sparse packing constraints, and obtain an e2ke−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.