ℓ₁ -Sparsity approximation bounds for packing integer programs

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,jb_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 ℓ₁-column sparsity of A (denoted by Δ₁) 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 Δ₁. 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 Δ₁≤ 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(ϵ²/ Δ₁) -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.