On k-Column Sparse Packing Programs

TL;DR

This paper presents an improved approximation algorithm for k-column sparse packing integer programs, achieving a better ratio than previous methods, and extends the results to submodular objectives, highlighting both algorithmic advances and theoretical limits.

Contribution

It introduces an (ek+o(k))-approximation algorithm for k-column sparse PIPs and extends it to submodular maximization, improving upon prior bounds.

Findings

Achieves an (ek+o(k))-approximation ratio for k-column sparse PIPs.

Shows the integrality gap of the LP relaxation is at least 2k-1.

Extends results to submodular maximization with similar approximation guarantees.

Abstract

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.