In this paper we propose an improved approximation scheme for the Vector Bin Packing problem (VBP), based on the combination of (near-)optimal solution of the Linear Programming (LP) relaxation and a greedy (modified first-fit) heuristic. The Vector Bin Packing problem of higher dimension (d 2) is not known to have asymptotic polynomial-time approximation schemes (unless P = NP). Our algorithm improves over the previously-known guarantee of (ln d + 1 + epsilon) by [1] for higher dimensions (d > 2). We provide a (1) approximation scheme for certain set of inputs for any dimension d. More precisely, we provide a 2-OPT algorithm, a result which is irrespective of the number of dimensions d.
One dimensional bin packing problem has been studied extensively. Fernandez de la Vega and Lueker @cite_0 gave the first asymptotic polynomial-time approximation scheme (APTAS). They put forward a rounding technique that allowed them to reduce the problem of packing large items to finding an optimum packing of just a constant number of items (at a cost of @math times OPT). Their algorithm was later improved by Karmarkar and Karp @cite_6 , to a (1+ @math )-OPT bound.
@cite_0 @cite_6
2067823599 2098705232 2123068254 2101057470
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. On-line algorithms have been extensively studied for the one-dimensional bin packing problem. In this paper, we investigate two classes of one-dimensional bin packing algorithms, and we give better lower bounds for their asymptotic worst-case behavior. For on-line algorithms so far the best lower bound was given by van Vliet in (1992) [12]. He proved that there is no on-line bin packing algorithm with better asymptotic performance ratio than 1.54014.... In this paper, we give an improvement on this bound to 248161=1.54037... and we investigate the parametric case as well. For those lists where the elements are preprocessed according to their sizes in non-increasing order, (1983) [1] proved that no on-line algorithm can have an asymptotic performance ratio smaller than 87. We improve this result to 5447. We present several polynomial-time approximation algorithms for the one-dimensional bin-packing problem. using a subroutine to solve a certain linear programming relaxation of the problem. Our main results are as follows: There is a polynomial-time algorithm A such that A(I) ≤ OPT(I) + O(log2 OPT(I)). There is a polynomial-time algorithm A such that, if m(I) denotes the number of distinct sizes of pieces occurring in instance I, then A(I) ≤ OPT(I) + O(log2 m(I)). There is an approximation scheme which accepts as input an instance I and a positive real number e, and produces as output a packing using as most (1 + e) OPT(I) + O(e-2) bins. Its execution time is O(e-c n log n), where c is a constant. These are the best asymptotic performance bounds that have been achieved to date for polynomial-time bin-packing. Each of our algorithms makes at most O(log n) calls on the LP relaxation subroutine and takes at most O(n log n) time for other operations. The LP relaxation of bin packing was solved efficiently in practice by Gilmore and Gomory. We prove its membership in P, despite the fact that it has an astronomically large number of variables. The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit.