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

A number of intriguing decision scenarios revolve around partitioning a collection 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 version of OPP, namely, the Stochastic On-line Equi-Partitioning Problem (SO-EPP). In SO-EPP, the target partitioning is unknown and has to be inferred purely from observing an on-line sequence of object pairs. The paired objects belong to the same partition with probability $p$ and to different partitions with probability $1-p$, with $p$ also being unknown. As an additional complication, the partitions are required to be of equal cardinality. Previously, only sub-optimal solution strategies have been proposed for SO- EPP. In this paper, we propose the first optimal solution strategy. In brief, the scheme that we propose, BN-EPP, is founded on a Bayesian network representation of SO-EPP problems. Based on probabilistic reasoning, we are not only able to infer the underlying object partitioning with optimal accuracy. We are also able to simultaneously infer $p$, allowing us to accelerate learning as object pairs arrive. Furthermore, our scheme is the first to support arbitrary constraints on the partitioning (Constrained SO-EPP). Being optimal, BN-EPP provides superior performance compared to existing solution schemes. We additionally introduce Walk-BN-EPP, a novel WalkSAT inspired algorithm for solving large scale BN-EPP problems. Finally, we provide a BN-EPP based solution to the problem of order picking, a representative real-life application of BN-EPP.