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Multi-dimensional classification (MDC) aims at finding a function that assigns a vector of class values to a vector of observed features. Multi-dimensional Bayesian network classifier (MBNC) was devised for MDC in 2006, but with restricted structure. By removing the constraints, an undocumented model called general multi-dimensional Bayesian network classifier (GMBNC) is proposed in this article, along with an exact induction algorithm which is able to recover the GMBNC by local search, without having to learn the whole BN first. We prove its soundness, and conduct experimental studies to verify its effectiveness and efficiency. The larger is the problem, the more saving by IPC-GMBNC versus conventional approach (global structure learning by PC algorithm), e.g. given an example network with 200 nodes, around 99% saving is achieved.