We consider column-sparse covering integer programs, a generalization of set
cover, which have a long line of research of (randomized) approximation
algorithms. We develop a new rounding scheme based on the Partial Resampling
variant of the Lov\'{a}sz Local Lemma developed by Harris & Srinivasan (2019).
This achieves an approximation ratio of 1+aminβln(Ξ1β+1)β+O(log(1+aminβlog(Ξ1β+1)ββ), where aminβ is the minimum covering
constraint and Ξ1β is the maximum β1β-norm of any column of the
covering matrix (whose entries are scaled to lie in [0,1]). When there are
additional constraints on the variable sizes, we show an approximation ratio of
lnΞ0β+O(loglogΞ0β) (where Ξ0β is the maximum number
of non-zero entries in any column of the covering matrix). These results
improve asymptotically, in several different ways, over results of Srinivasan
(2006) and Kolliopoulos & Young (2005).
We show nearly-matching inapproximability and integrality-gap lower bounds.
We also show that the rounding process leads to negative correlation among the
variables, which allows us to handle multi-criteria programs