The problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles is known to be NP-hard. In this paper, we first develop a 4-approximation algorithm for the special case of the problem in which P is a 3D histogram. It runs in O(mlogm)\documentclass[12pt]{minimal}
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\begin{document}$$O(m \log m)$$\end{document} time, where m is the number of corners in P. We then apply it to exactly compute the arithmetic matrix product of two n×n\documentclass[12pt]{minimal}
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\begin{document}$$n \times n$$\end{document} matrices A and B with nonnegative integer entries, yielding a method for computing A×B\documentclass[12pt]{minimal}
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\begin{document}$$A \times B$$\end{document} in O~(n2+min{rArB,nmin{rA,rB}})\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{O}(n^2 + \min \{ r_A r_B,\, n \min \{r_A,\ r_B\}\})$$\end{document} time, where O~\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{O}$$\end{document} suppresses polylogarithmic (in n) factors and where rA\documentclass[12pt]{minimal}
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\begin{document}$$r_A$$\end{document} and rB\documentclass[12pt]{minimal}
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\begin{document}$$r_B$$\end{document} denote the minimum number of 3D rectangles into which the 3D histograms induced by A and B can be partitioned, respectively.
