We investigate how to partition a rectangular region of length @math and height @math into @math rectangles of given areas @math using two-stage guillotine cuts, so as to minimize either (i) the sum of the perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of the rectangles. These problems play an important role in the ongoing Vietnamese land-allocation reform, as well as in the optimization of matrix multiplication algorithms. We show that the first problem can be solved to optimality in @math , while the two others are NP-hard. We propose mixed integer programming (MIP) formulations and a binary search-based approach for solving the NP-hard problems. Experimental analyses are conducted to compare the solution approaches in terms of computational efficiency and solution quality, for different objectives.
@cite_5: We address the issue of implementing matrix multiplication on heterogeneous platforms. We target two different classes of heterogeneous computing resources: heterogeneous networks of workstations and collections of heterogeneous clusters. Intuitively, the problem is to load balance the work with different speed resources while minimizing the communication volume. We formally state this problem in a geometric framework and prove its NP-completeness. Next, we introduce a (polynomial) column-based heuristic, which turns out to be very satisfactory: We derive a theoretical performance guarantee for the heuristic and we assess its practical usefulness through MPI experiments.
@cite_6: 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 s 1 , s 2 , . . . ,s p (such that Σ i=1 p s i = 1 ), so as to minimize either (i) the sum of the p perimeters of the rectangles or (ii) the largest perimeter of the p rectangles? For both problems, we prove NP-completeness and we introduce a 7 4 -approximation algorithm for (i) and a @math -approximation algorithm for (ii).
Partitioning an area into polygons of fixed shape or area is a class of problems which has been regularly studied in the operations research and computational geometry literature. @cite_5 @cite_6 defined two optimization problems that seek to partition the unit square into a number of rectangles with given areas, so as to optimize parallel matrix-multiplication algorithms in heterogeneous parallel computing platforms. The first problem aims to minimize the sum of all rectangle perimeters, whereas the second problem aims to minimize the largest perimeter. These problems are special cases of and where the general rectangular region is a square. The authors introduced an @math -approximate algorithm and an @math -approximate algorithm for these problems, respectively.