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.
Other close variants of have been studied. @cite_0 @cite_4 considered the problem of decomposing a square or a rectangle into a number of rectangles of equal area so as to minimize the maximum rectangle perimeter. VLSI floorplan design and facility location applications also led to a number of related studies . @cite_2 proposed a local search and mathematical programming algorithm to solve rectangular packing problems where the shapes of the rectangles are adjustable within given perimeter and area constraints.
@cite_0 @cite_4 @cite_2
2060963575 1967786783 2067823599 2024500486
We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classicalNP-hard cutting-stock problem. The algorithm, based on a new linear-programming relaxation, finds a packing ofn rectangles whose total height is within a factor of (1 +e) of optimal (up to an additive term), and has running time polynomial both in n and in 1 e. The Steiner packing problem is to find the maximum number of edge-disjoint subgraphs of a given graph G that connect a given set of required points S. This problem is motivated by practical applications in VLSI- layout and broadcasting, as well as theoretical reasons. In this paper, we study this problem and present an algorithm with an asymptotic approximation factor of vSv 4. This gives a sufficient condition for the existence of k edge-disjoint Steiner trees in a graph in terms of the edge-connectivity of the graph. We will show that this condition is the best possible if the number of terminals is 3. At the end, we consider the fractional version of this problem, and observe that it can be reduced to the minimum Steiner tree problem via the ellipsoid algorithm. We study the following packing problem: Given a collection of d-dimensional rectangles of specified sizes, pack them into the minimum number of unit cubes. We show that unlike the one-dimensional case, the two-dimensional packing problem cannot have an asymptotic polynomial time approximation scheme (APTAS), unless PNP. On the positive side, we give an APTAS for the special case of packing d-dimensional cubes into the minimum number of unit cubes. Second, we give a polynomial time algorithm for packing arbitrary two-dimensional rectangles into at most OPT square bins with sides of length 1 , where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result has no additive constant term, i.e., is not an asymptotic result. As a corollary, we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum-area encasing rectangle. Given a set S of n points in the plane, the disjoint two-rectangle covering problem is to find a pair of disjoint rectangles such that their union contains S and the area of the larger rectangle is minimized. In this paper we consider two variants of this optimization problem: (1) the rectangles are allowed to be reoriented freely while restricting them to be parallel to each other, and (2) one rectangle is restricted to be axis-parallel but the other rectangle is allowed to be reoriented freely. For both of the problems, we present O(n2log n)-time algorithms using O(n) space.