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.
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.
@cite_5 @cite_6
1494149077 2022366735 1688647404 1998375208
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). Problems concerned with finding inscribing or circumscribing polygons that maximize some measurement are considered such as: Find an area maximizing triangle inscribed in a given convex polygon. Algorithms solving a number of these problems in linear time are presented. They use the common approach of finding an initial solution with respect to a fixed bounding point and then iteratively transforming this solution into a new solution with respect to a new point. The generality of this approach is discussed and several open problems are noted. We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O ( n 3 ) . We also give ( 1 - e ) -approximation algorithms that take time O ( e - 1 2 log ? n + e - 3 2 ) . In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1 + e)-approximation algorithm for the MWISR problem with quasipolynomial running time 2poly(log n e). In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to [4] this allows us to upper bound our approximation ratio by 1 + e. Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1 + e)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem. In particular, note that our results imply that the MWISR problem is not APX-hard, unless NP ⊆ DTIME(2polylog (n)).