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A Multi-Phase Search Approach to the LEGO Construction Problem

Last modified: 2016-06-20

#### Abstract

The task of determining which LEGO bricks to use to construct a volume is known as the LEGO Construction Problem. This is a challenging problem because even small volumes can be constructed in a tremendously large number of ways. As a result, an exhaustive search is impractical, and more nuanced search strategies must be employed to find a good, though not necessarily optimal, solution. This paper describes a multi-phase search approach to the LEGO Construction Problem. Our first search phase uses heuristics to identify a moderate number of candidates for each layer in the model. This is followed by two different search strategies which identify alternative brick arrangements that reduce the number of connected components, undesirable edges, and bricks in the model. A final highly localized search is applied to bricks at the boundaries between the model's connected components if the previous search processes fail to reduce the model to a single connected component. Applying this four-phase search strategy to a diverse selection of models has demonstrated that it normally finds a result that consists of a single connected component when such a solution exists, and that the models are structurally sound when built.

#### Keywords

LEGO Construction Problem; Brick Layout Problem; Local Search; Multi-Phase Search; LEGO

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