Bin Packing and Covering: Pushing the Frontier on the Maximin Share Fairness
Bo Li, Ankang Sun, Zunyu Wang, Yu Zhou
TL;DR
This work studies MMS-based fair allocation for two canonical combinatorial problems: bin covering (goods) and bin packing (chores). It introduces two relaxations—cardinal-approximation and ordinal-approximation—and proves constant-factor guarantees for each model, with $2/3$-CMMS and $3/4$-OMMS in bin covering and $4/3$-CMMS and $4/3$-OMMS (plus small additive loss) in bin packing. The main technical contributions include a constrained partition-and-match approach with envy-free matchings for the cardinal case and a Round-Robin-like scheme with synthesized items for the ordinal case, together with structural lemmas that bound how items can be combined into bins. The results extend MMS fairness to non-atomic group settings and provide efficient algorithms that operate on identical-ordering (IDO) instances, offering practical implications for resource allocation in cloud, logistics, and related domains. Overall, the paper advances the frontier of MMS-inspired fair division in group-centric, combinatorial environments by delivering scalable, provably fair allocation procedures under two natural dual models.
Abstract
We study a fundamental fair allocation problem, where the agent's value is determined by the number of bins either used to pack or cover the items allocated to them. Fairness is evaluated using the maximin share (MMS) criterion. This problem is not only motivated by practical applications, but also serves as a natural framework for studying group fairness. As MMS is not always satisfiable, we consider two types of approximations: cardinal and ordinal. For cardinal approximation, we relax the requirements of being packed or covered for a bin, and for ordinal approximation, we relax the number of bins that are packed or covered. For all models of interest, we provide constant approximation algorithms.