A study on load-balanced variants of the bin packing problem

TL;DR

This paper explores load-balanced variants of the fractional bin packing problem, introducing new constraints, proving NP-hardness for certain cases, and proposing an optimal polynomial-time algorithm for a specific multi-round allocation scenario.

Contribution

It formulates load-balanced fractional bin packing variants, proves NP-hardness for limited splits, and provides an optimal polynomial-time solution for multi-round allocations.

Findings

NP-hardness when objects are split across at most 3 bins

Two rounds suffice for complete object assignment in the multi-round model

An optimal polynomial-time algorithm for the multi-round load-balanced allocation

Abstract

We consider several extensions of the fractional bin packing problem, a relaxation of the traditional bin packing problem where the objects may be split across multiple bins. In these extensions, we introduce load-balancing constraints imposing that the share of each object which is assigned to a same bin must be equal. We propose a Mixed-Integer Programming (MIP) formulation and show that the problem becomes NP-hard if we limit to at most 3 the number of bins across which each object can be split. We then consider a variant where the balanced allocations of objects to bins may be done in successive rounds; this problem was inspired by telecommunication applications, and may be used to model simple Live Streaming networks. We show that two rounds are always sufficient to completely assign all objects to the bins and then provide an optimal polynomial-time allocation algorithm for this problem.