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A number of intriguing decision scenarios revolve around partitioning acollection of objects to optimize some application specific objective function.This problem is generally referred to as the Object Partitioning Problem (OPP)and is known to be NP-hard. We here consider a particularly challenging versionof OPP, namely, the Stochastic On-line Equi-Partitioning Problem (SO-EPP). InSO-EPP, the target partitioning is unknown and has to be inferred purely fromobserving an on-line sequence of object pairs. The paired objects belong to thesame partition with probability $p$ and to different partitions withprobability $1-p$, with $p$ also being unknown. As an additional complication,the partitions are required to be of equal cardinality. Previously, onlysub-optimal solution strategies have been proposed for SO- EPP. In this paper,we propose the first optimal solution strategy. In brief, the scheme that wepropose, BN-EPP, is founded on a Bayesian network representation of SO-EPPproblems. Based on probabilistic reasoning, we are not only able to infer theunderlying object partitioning with optimal accuracy. We are also able tosimultaneously infer $p$, allowing us to accelerate learning as object pairsarrive. Furthermore, our scheme is the first to support arbitrary constraintson the partitioning (Constrained SO-EPP). Being optimal, BN-EPP providessuperior performance compared to existing solution schemes. We additionallyintroduce Walk-BN-EPP, a novel WalkSAT inspired algorithm for solving largescale BN-EPP problems. Finally, we provide a BN-EPP based solution to theproblem of order picking, a representative real-life application of BN-EPP.

An Optimal Bayesian Network Based Solution Scheme for the Constrained Stochastic On-line Equi-Partitioning Problem