Partial Resampling to Approximate Covering Integer Programs

We consider column-sparse positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (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 (2013). This achieves an approximation ratio of $1 + \frac{\ln (\Delta_1+1)}{a_{\text{min}}} + O(\sqrt{ \frac{\log (\Delta_1+1)}{a_{\text{min}}}} )$, where $a_{\text{min}}$ is the minimum covering constraint and $\Delta_1$ is the maximum $\ell_1$-norm of any column of the covering matrix (whose entries are scaled to lie in $[0,1]$). When there are additional constraints on the sizes of the variables, we show an approximation ratio of $1 + O(\frac{\ln (\Delta_1+1)}{a_{\text{min}} \epsilon} + \sqrt{ \frac{\log (\Delta_1+1)}{a_{\text{min}}}})$ to satisfy these size constraints up to multiplicative factor $1 + \epsilon$, or an approximation of ratio of $\ln \Delta_0 + O(\sqrt{\log \Delta_0})$ to satisfy the size constraints exactly (where $\Delta_0$ is the maximum number of non-zero entries in any column of the covering matrix). We also show nearly-matching inapproximability and integrality-gap lower bounds. These results improve asymptotically, in several different ways, over results shown by Srinivasan (2006) and Kolliopoulos \& Young (2005). We show also that our algorithm automatically handles multi-criteria programs, efficiently achieving approximation ratios which are essentially equivalent to the single-criterion case and which apply even when the number of criteria is large.