English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider approximation algorithms for packing integer programs (PIPs) of the form $max{langle c, xrangle : Ax le b, x  {0,1}^n}$ 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. cite{bansal-sparse} obtained an $Omega(frac{1}{Delta_0^{1/lfloor W rfloor}})$-approximation ratio where $Delta_0$ 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 $Delta_1$) which can be much smaller than $Delta_0$. Motivated by recent work on covering integer programs (CIPs) cite{cq,chs-16} we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. cite{bansal-sparse} (but with a twist), yield approximation ratios for PIPs based on $Delta_1$. First, following an integrality gap example from cite{bansal-sparse}, we observe that the case of $W=1$ is as hard as maximum independent set even when $Delta_1 le 2$. 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 $epsilon  (0,1]$, we obtain an $Omega(epsilon^2/Delta_1)$-approximation. In the large width regime, when $W ge 2$, we obtain an $Omega((frac{1}{1 + Delta_1/W})^{1/(W-1)})$-approximation. We also obtain a $(1-epsilon)$-approximation when $W = Omega(frac{log (Delta_1/epsilon)}{epsilon^2})$.

$$\ell _1$$ -sparsity Approximation Bounds for Packing Integer Programs