Partial Resampling to Approximate Covering Integer Programs

Abstract

We consider column-sparse covering integer programs, a generalization of set cover, 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 +, where a min is the minimum covering constraint and ∆ 1 is the maximum ℓ 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 log(∆1+1) aminǫ + log(∆1+1) amin to satisfy these size constraints up to multiplicative factor 1 + ǫ, or an approximation 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 the rounding process leads to negative correlation among the variables. This allows us to automatically handle multi-criteria programs, efficiently achieving approximation ratios which are essentially equivalent to the single-criterion case and apply even when the number of criteria is large.

On Approximating (Sparse) Covering Integer Programs

“…The first issue is regarding the approximation ratios for CIP and CIP ∞ . Currently the best bounds in terms of column sparsity are from the recent work of Chen, Harris, and Srinivasan [12]. To describe the bounds, we assume, without loss of generality, that the problem is normalized such that entries of A are in [0, 1] and b ≥ 1.…”

“…To describe the bounds, we assume, without loss of generality, that the problem is normalized such that entries of A are in [0, 1] and b ≥ 1. Following [12], we let ∆ 0 denote the maximum number of non-zeroes in any column of A, and let ∆ 1 = max…”

“…As a byproduct of the simplicity of the alteration algorithm and analysis, we can derandomize the algorithm without any loss in the approximation guarantee or efficiency. Previous work by Chen, Harris and Srinivasan [12] which obtained near-tight bounds is based on a resampling-based randomized algorithm whose analysis is complex.Non-trivial approximation algorithms for CIP are based on solving the natural LP relaxation strengthened with knapsack cover (KC) inequalities [5,25,12]. Our second contribution is a fast (essentially near-linear time) approximation scheme for solving the strengthened LP with a factor of n speed up over the previous best running time [5].…”

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

“…Our scheme is similar to that of Bansal et al at a high level but we make a simple but important change in the algorithm and its analysis. This is inspired by recent work on covering integer programs [4] where ℓ 1 -sparsity based approximation bounds from [6] were simplified.…”

“…They raised the question of obtaining approximation ratios based on the ℓ1-column sparsity of A (denoted by ∆1) which can be much smaller than ∆0. Motivated by recent work on covering integer programs (CIPs) [4,6] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al [1] (but with a twist), yield approximation ratios for PIPs based on ∆1. First, following an integrality gap example from [1], we observe that the case of W = 1 is as hard as maximum independent set even when ∆1 ≤ 2.…”