Heterogeneous matrix-matrix multiplication or partitioning a square into rectangles: NP-completeness and approximation algorithms

In this paper, we deal with two geometric problems arising from heterogeneous parallel computing: how to partition the unit square into p rectangles of given area s₁, s₂, ..., s_p (such that Σ_i=1^p s_i=1), so as to minimize (i) either the sum of the p perimeters of the rectangles (ii) or the largest perimeter of the p rectangles. For both problems, we prove NP-completeness and we introduce approximation algorithms