An Approximation Algorithm for Covering Linear Programs and its Application to Bin-Packing

TL;DR

This paper presents a modified approximation algorithm for covering linear programs that extends to complex bin-packing problems, providing solutions even when the associated knapsack problem is poorly approximated.

Contribution

It introduces a simple modification of the Plotkin-Shmoys-Tardos algorithm to achieve approximation for covering LPs under broader conditions, applicable to large classes of bin-packing problems.

Findings

Achieves an lpha(1+psilon)-approximation for covering LPs.

Extends applicability to bin-packing problems with poorly approximated knapsack subproblems.

Works even when the knapsack problem has a large approximation factor.

Abstract

We give an $\alpha(1+\epsilon)$-approximation algorithm for solving covering LPs, assuming the presence of a $(1/\alpha)$-approximation algorithm for a certain optimization problem. Our algorithm is based on a simple modification of the Plotkin-Shmoys-Tardos algorithm (MOR 1995). We then apply our algorithm to $\alpha(1+\epsilon)$-approximately solve the configuration LP for a large class of bin-packing problems, assuming the presence of a $(1/\alpha)$-approximate algorithm for the corresponding knapsack problem (KS). Previous results give us a PTAS for the configuration LP using a PTAS for KS. Those results don't extend to the case where KS is poorly approximated. Our algorithm, however, works even for polynomially-large $\alpha$.