A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Abstract

In this paper we develop new techniques for covering linear programs and covering integer linear programs 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. [Math. Programming, 34 (1986), pp. 251--264] in the sensitivity analysis for linear programs and integer linear programs. This result is often applied in the design of robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign already packed items. The migration factor measures the amount of repacked items. It 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 $\frac{1}{\epsilon}$. To the best of our knowledge, this is the first (asymptotic) PTAS with polynomial migration for an NP-hard problem. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook et al. [Math. Programming, 34 (1986), pp. 251--264] for linear programs.