Algorithms to Approximate Column-Sparse Packing Problems

TL;DR

This paper introduces new algorithms using attenuation and multiple-chance techniques to improve approximation ratios for column-sparse packing problems, achieving near-optimal solutions for several well-studied cases.

Contribution

It presents unifying ideas that enhance approximation algorithms for column-sparse packing problems, including LP relaxation bounds and stochastic variants.

Findings

Achieved the integrality gap for k-column sparse packing integer programs.

Improved approximation for stochastic k-set packing.

Approached half the gap for a hypergraph matching conjecture.

Abstract

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).