$\ell _1$-Sparsity approximation bounds for packing integer programs

Abstract

We consider approximation algorithms for packing integer programs (PIPs) of the form $\max \{\langle c, x\rangle : Ax \le b, x \in \{0,1\}^n\}$ 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 $\varOmega (\frac{1}{\varDelta _0^{1/\lfloor W \rfloor }})$-approximation ratio where $\varDelta _0$ is the maximum number of nonzeroes in any column of A (in other words the $\ell _0$-column sparsity of A). They raised the question of obtaining approximation ratios based on the $\ell _1$-column sparsity of A (denoted by $\varDelta _1$) which can be much smaller than $\varDelta _0$. 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 $\varDelta _1$. 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 $\varDelta _1 \le 2$. 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 $W = 1 + \epsilon$ where $\epsilon \in (0,1]$, we obtain an $\varOmega (\epsilon ^2/\varDelta _1)$-approximation. In the large width regime, when $W \ge 2$, we obtain an $\varOmega ((\frac{1}{1 + \varDelta _1/W})^{1/(W-1)})$-approximation. We also obtain a $(1-\epsilon )$-approximation when $W = \varOmega (\frac{\log (\varDelta _1/\epsilon )}{\epsilon ^2})$. 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.