### Abstract

We consider approximation algorithms for packing integer programs (PIPs) of the form max { ⟨ c, x⟩ : Ax≤ b, x∈ { 0 , 1 } ^{n}} where A, b and c are nonnegative. We let W= min _{i} _{,} _{j}b_{i}/ A_{i} _{,} _{j} denote the width of A which is at least 1. Previous work by Bansal et al. (Theory Comput 8(24):533–565, 2012) obtained an Ω(1Δ01/⌊W⌋)-approximation ratio where Δ is the maximum number of nonzeroes in any column of A (in other words the ℓ-column sparsity of A). They raised the question of obtaining approximation ratios based on the ℓ_{1}-column sparsity of A (denoted by Δ_{1}) which can be much smaller than Δ. Motivated by recent work on covering integer programs (Chekuri and Quanrud, in: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp 1596–1615. SIAM, 2019; Chen et al., in: Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp 1984–2003. SIAM, 2016) we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. (Theory Comput 8(24):533–565, 2012) (but with a twist), yield approximation ratios for PIPs based on Δ_{1}. First, following an integrality gap example from (Theory Comput 8(24):533–565, 2012), we observe that the case of W= 1 is as hard as maximum independent set even when Δ_{1}≤ 2. 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 W= 1 + ϵ where ϵ∈ (0 , 1] , we obtain an Ω(ϵ^{2}/ Δ_{1}) -approximation. In the large width regime, when W≥ 2 , we obtain an Ω((11+Δ1/W)1/(W-1))-approximation. We also obtain a (1 - ϵ) -approximation when W=Ω(log(Δ1/ϵ)ϵ2). Viewing the rounding algorithms as contention resolution schemes, we obtain approximation algorithms in the more general setting when the objective is a non-negative submodular function.

Original language | English (US) |
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Journal | Mathematical Programming |

DOIs | |

State | Accepted/In press - Jan 1 2020 |

### Keywords

- Approximation algorithms
- Randomized rounding
- Sparse packing integer programs
- Submodular optimization

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

_{1}-Sparsity approximation bounds for packing integer programs.

*Mathematical Programming*. https://doi.org/10.1007/s10107-020-01472-7