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A Bayesian network based solution scheme for the constrained Stochastic On-line Equi-Partitioning Problem

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Abstract

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 heuristic sub-optimal solution strategies have been proposed for SO- EPP. In this paper, we propose the first Bayesian 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 superior 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 a wide range of constraints on the partitioning (Constrained SO-EPP). Being Bayesian, 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.

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Notes

  1. Note that the arrival of objects in pairs can easily be generalized to arrival of objects in sets, which in turn are transformed into pairwise combinations of the objects contained in each set. See Section 5.4 for an example of this.

  2. These are based on real-world point-of-sale transaction data from a grocery outlet [3].

  3. Also known as Most Probable Explanation (MPE).

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Authors and Affiliations

  1. Center for Artificial Intelligence Research (CAIR), University of Agder, Postbox 422, 4604, Kristiansand, Norway

    Sondre Glimsdal & Ole-Christoffer Granmo

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  1. Sondre Glimsdal
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  2. Ole-Christoffer Granmo
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Correspondence to Sondre Glimsdal.

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Glimsdal, S., Granmo, OC. A Bayesian network based solution scheme for the constrained Stochastic On-line Equi-Partitioning Problem. Appl Intell 48, 3735–3747 (2018). https://doi.org/10.1007/s10489-018-1172-8

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