A Robust AFPTAS for Online Bin Packing with Polynomial Migration

TL;DR

This paper introduces a new AFPTAS for online bin packing that allows limited item reassignments, achieving near-optimal solutions with polynomial migration, and extends sensitivity analysis for LPs and ILPs.

Contribution

Develops a robust AFPTAS for online bin packing with polynomial migration, answering an open question and extending sensitivity analysis for LPs and ILPs.

Findings

Achieves an AFPTAS with polynomial migration bound.

Provides an approximate sensitivity theorem for LPs and ILPs.

Answers an open question by Epstein and Levin.

Abstract

In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is often applied in designing robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign a certain number of items, measured by the migration factor. The migration factor is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in $\frac{1}{\epsilon}$. This answers an open question stated by Epstein and Levin in the affirmative. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook at el. for linear programs.