### Abstract

We consider approximation algorithms for packing integer programs (PIPs) of the form where c, A, and b are nonnegative. We let denote the width of A which is at least 1. Previous work by Bansal et al. [1] obtained an -approximation ratio where is the maximum number of nonzeroes in any column of A (in other words the -column sparsity of A). They raised the question of obtaining approximation ratios based on the -column sparsity of A (denoted by) which can be much smaller than Motivated by recent work on covering integer programs (CIPs) [4, 7] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. [1] (but with a twist), yield approximation ratios for PIPs based on First, following an integrality gap example from [1], we observe that the case of is as hard as maximum independent set even when In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width where we obtain an -approximation. In the large width regime, when we obtain an -approximation. We also obtain a -approximation when.

Original language | English (US) |
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Title of host publication | Integer Programming and Combinatorial Optimization - 20th International Conference, IPCO 2019, Proceedings |

Editors | Andrea Lodi, Viswanath Nagarajan |

Publisher | Springer-Verlag Berlin Heidelberg |

Pages | 128-140 |

Number of pages | 13 |

ISBN (Print) | 9783030179526 |

DOIs | |

State | Published - Jan 1 2019 |

Event | 20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019 - Ann Arbor, United States Duration: May 22 2019 → May 24 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11480 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019 |
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Country | United States |

City | Ann Arbor |

Period | 5/22/19 → 5/24/19 |

### Keywords

- Approximation algorithms
- Packing integer programs
- column sparsity

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Integer Programming and Combinatorial Optimization - 20th International Conference, IPCO 2019, Proceedings*(pp. 128-140). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11480 LNCS). Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-030-17953-3_10