Partial resampling to approximate covering integer programs

TL;DR

This paper introduces a new partial resampling rounding scheme for column-sparse covering integer programs, achieving improved approximation ratios and handling multi-criteria constraints, advancing the state of approximation algorithms.

Contribution

It develops a novel rounding scheme based on the Partial Resampling Lovász Local Lemma, improving approximation ratios for covering integer programs with various constraints.

Findings

Achieves approximation ratio of 1 + (ln(Δ₁+1))/a_min + O(log(1 + sqrt(ln(Δ₁+1)/a_min))).

Provides approximation ratio of ln Δ₀ + O(log log Δ₀) with additional variable size constraints.

Establishes nearly-matching inapproximability and integrality-gap lower bounds.

Abstract

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 $1 + \frac{\ln (\Delta_1+1)}{a_{\min}} + O\Big( \log(1 + \sqrt{ \frac{\log (\Delta_1+1)}{a_{\min}}} \Big)$, where $a_{\min}$ 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 variable sizes, we show an approximation ratio of $\ln \Delta_0 + O(\log \log \Delta_0)$ (where $\Delta_0$ 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.