$$\ell _1$$-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 c, A, and b 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. [1] 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 (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 \(\varDelta _1\). First, following an integrality gap example from [1], 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 \(W = \varOmega (\frac{\log (\varDelta _1/\epsilon )}{\epsilon ^2})\).