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 Lovász Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln(∆1+1) amin +O( √ log(∆1+1) amin ), where amin is the minimum covering constraint and ∆1 is the maximum l1-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( log(∆1+1) aminǫ + √ log(∆1+1) amin ) to satisfy these size constraints up to multiplicative factor 1 + ǫ, or an approximation of ratio of ln∆0 +O( √ log∆0) to satisfy the size constraints exactly (where ∆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.