This paper focuses on the description and computation of the B-differential of the componentwise minimum of two affine vector functions. This issue arises in the reformulation of the linear complementarity problem with the Min C-function. The question has many equivalent formulations and we identify some of them in linear algebra, convex analysis and discrete geometry. These formulations are used to state some properties of the B-differential, like its symmetry, condition for its completeness, its connectivity, bounds on its cardinal, etc. The set to specify has a finite number of elements, which may grow exponentially with the range space dimension of the functions, so that its description is most often algorithmic. We present first an incremental-recursive approach avoiding to solve any optimization subproblem, which is based on the notion of matroid circuit and the related introduced concept of stem vector. Next, we propose modifications, adapted to the problem at stake, of an algorithm introduced by Rada and Černý in 2018 for determining the cells of an arrangement in the space of hyperplanes having a point in common. Measured in CPU time on the considered test-problems, the mean acceleration ratios of the proposed algorithms, with respect to the one of Rada and Černý, are in the range 7..15, and this speed-up can exceed 30, depending on the problem and the approach.

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