An AFPTAS for Bin Packing with Partition Matroid via a New Method for LP Rounding

We consider the Bin Packing problem with a partition matroid constraint. The input is a set of items of sizes in [0 , 1], and a partition matroid over the items. The goal is to pack the items in a minimum number of unit-size bins, such that each bin forms an independent set in the matroid. This variant of classic Bin Packing has natural applications in secure storage on the Cloud, as well as in equitable scheduling and clustering with fairness constraints. Our main result is an asymptotic fully polynomial-time approximation scheme (AFPTAS) for Bin Packing with a partition matroid constraint. This scheme generalizes the known AFPTAS for Bin Packing with Cardinality Constraints and improves the existing asymptotic polynomial-time approximation scheme (APTAS) for Group Bin Packing, which are both special cases of Bin Packing with partition matroid. We derive the scheme via a new method for rounding a (fractional) solution for a configuration-LP. Our method uses this solution to obtain prototypes , in which items are interpreted as placeholders for other items, and applies fractional grouping to modify a fractional solution (prototype) into one having desired integrality properties