Selfish Square Packing

We consider a game-theoretical bin packing problem. The 1D (one dimensional) case has been treated in the literature as the ʼselfish bin packing problemʼ. We investigate a 2D version, in which the items to be packed are squares and the bins are unit squares. In this game, a set of items is packed into bins. Each player controls exactly one item and is charged with a cost defined as the ratio between the area of the item and the occupied area of the respective bin. One at a time, players selfishly move their items from one bin to another, in order to minimize the costs they are charged. At a Nash equilibrium, no player can reduce the cost he is charged by moving his item to a different bin. In the 2D case, to decide whether an item can be placed in another bin with other items is NP-complete, so we consider that players use a packing algorithm to make this decision. We show that this game converges to a Nash equilibrium, independently of the packing algorithm used. We prove that the price of anarchy is at least 2.27. We also prove that, using the NFDH packing algorithm, the asymptotic price of anarchy is at most 2.6875.     

An asymptotic approximation scheme for the concave cost bin packing problem

We consider a generalized one-dimensional bin packing model in which the cost of a bin is a nondecreasing concave function of the utilization of the bin. We show that for any given positive constant ?, there exists a polynomial-time approximation algorithm with an asymptotic worst-case performance ratio of no more than 1 + ?.     

带参量的非合作装箱博弈

带参量的非合作装箱博弈是指:每个物品的尺寸都介于0和参量x(0相似文献   

A note on a variant of the online open end bin packing problem

We study the minimum variant of the online open end bin packing problem. Items are presented one by one, and an item can be packed into a bin while the resulting total size of items excluding the minimum size item of the bin will be below 1. We design an improved online algorithm whose asymptotic competitive ratio does not exceed 1.180952381, and we prove a new lower bound of 1.1666666 on the asymptotic competitive ratio of any online algorithm.     

Packing items to feed assembly lines

We treat a practical application of packing problems in feeding assembly lines. This study was motivated by a real situation encountered in the shop floor of a major automobile industry plant in Brazil. The assembly line feed problem (LFP) consists in how pack the items in the available containers to meet the line work centers’ requirements with a minimum total cost over the planning horizon. LFP is a variable-sized bin packing problem that has two special features: (i) a cardinality constraint on each bin’s size; and, (ii) a cost structure such that each bin’s cost varies according to the items that are packed in it. We propose an integer programming model and a GRASP heuristic for LFP. Numerical results on real-life test instances are reported.     

Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.     

On-line bin packing — A restricted survey

In the classical bin packing problem, one is asked to pack items of various sizes into the minimum number of equal-sized bins. In the on-line version of this problem, the packer is given the items one by one and must immediately and irrevocably assign every item to its bin, without knowing the future items. Beginning with the first results in the early 1970's, we survey — from the worst case point of view — the approximation results obtained for on-line bin packing, higher dimensional versions of the problem, lower bounds on worst case ratios and related results.This work was partially supported by the Christian Doppier Laboratorium für Diskrete Optimierung.     

On the sum minimization version of the online bin covering problem

Given a set of m identical bins of size 1, the online input consists of a (potentially infinite) stream of items in (0,1]. Each item is to be assigned to a bin upon arrival. The goal is to cover all bins, that is, to reach a situation where a total size of items of at least 1 is assigned to each bin. The cost of an algorithm is the sum of all used items at the moment when the goal is first fulfilled. We consider three variants of the problem, the online problem, where there is no restriction of the input items, and the two semi-online models, where the items arrive sorted by size, that is, either by non-decreasing size or by non-increasing size. The offline problem is considered as well.     

The stochastic generalized bin packing problem

The Generalized Bin Packing Problem (GBPP) is a recently introduced packing problem where, given a set of bins characterized by volume and cost and a set of items characterized by volume and profit (which also depends on bins), we want to select a subset of items to be loaded into a subset of bins which maximizes the total net profit, while satisfying the volume and bin availability constraints. The total net profit is given by the difference between the total profit of the loaded items and the total cost of the used bins. In this paper, we consider the stochastic version of the GBPP (S-GBPP), where the item profits are random variables to take into account the profit oscillations due to the handling operations for bin loading. The probability distribution of these random variables is assumed to be unknown. By using the asymptotic theory of extreme values a deterministic approximation for the S-GBPP is derived.     

An approximation scheme for the two-stage,two-dimensional knapsack problem

We present an approximation scheme for the two-dimensional version of the knapsack problem which requires packing a maximum-area set of rectangles in a unit square bin, with the further restrictions that packing must be orthogonal without rotations and done in two stages. Achieving a solution which is close to the optimum modulo a small additive constant can be done by taking wide inspiration from an existing asymptotic approximation scheme for two-stage two-dimensional bin packing. On the other hand, getting rid of the additive constant to achieve a canonical approximation scheme appears to be widely nontrivial.     

一类带多重核元集在线装箱问题

本文讨论了一类在线变尺寸装箱问题,假定箱子的尺寸可以是不同的.箱子是在线到达的,仅当箱子到达后其尺寸才知道.给定一个带有核元的物品表及其上的核元关系图.我们的目标是要将表中元素装入到达的箱子中,保证任何箱子所装物品不互为核元,即所装物品对应的点所导出的子图是个空图,并使得所用的箱子总长最小.我们证明了该问题是NPHard的,并给出了基于图的点染色、图的团分解和基于背包问题的近似算法,给出了算法的时间复杂度和性能界.     

A note on a selfish bin packing problem

In this paper, we consider a selfish bin packing problem, where each item is a selfish player and wants to minimize its cost. In our new model, if there are k items packed in the same bin, then each item pays a cost 1/k, where k ≥ 1. First we find a Nash Equilibrium (NE) in time O(n log n) within a social cost at most 1.69103OPT + 3, where OPT is the social cost of an optimal packing; where n is the number of items or players; then we give tight bounds for the worst NE on the social cost; finally we show that any feasible packing can be converged to a Nash Equilibrium in O(n ²) steps without increasing the social cost.     

A note on the minimum bounded edge-partition of a tree

Minimum bounded edge-partition divides the edge set of a tree into the minimum number of disjoint connected components given a maximum weight for any component. It is an adaptation of the uniform edge-partition of a tree. An optimization algorithm is developed for this NP-hard problem, based on repeated bin packing of inter-related instances. The algorithm has linear running time for the class of ‘balanced trees’ common for the stochastic programming application which motivated investigation of this problem.Fast 2-approximation algorithms are formed for general instances by replacing the optimal bin packing with almost any bin packing heuristic. The asymptotic worst-case ratio of these approximation algorithms is never better than the absolute worst-case ratio of the bin packing heuristic used.     

Two-dimensional packing problems: A survey

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. We discuss mathematical models, and survey lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches. The relevant special cases where the items have to be packed into rows forming levels are also discussed in detail.     

An exact algorithm for the type-constrained and variable sized bin packing problem

In this paper, we introduce an additional constraint to the one-dimensional variable sized bin packing problem. Practically, some of items have to be packed separately in different bins due to their specific requirement. Therefore, these items are labelled as different types. The bins can be used to pack either any type of items if they are empty originally or the same type of items as what they already have. We model the problem as a type-constrained and variable sized bin packing problem (TVSBPP), and solve it via a branch and bound method. An efficient backtracking procedure is proposed to improve the efficiency of the algorithm.     

New classes of fast lower bounds for bin packing problems

The bin packing problem is one of the classical NP-hard optimization problems. In this paper, we present a simple generic approach for obtaining new fast lower bounds, based on dual feasible functions. Worst-case analysis as well as computational results show that one of our classes clearly outperforms the previous best “economical” lower bound for the bin packing problem by Martello and Toth, which can be understood as a special case. In particular, we prove an asymptotic worst-case performance of 3/4 for a bound that can be computed in linear time for items sorted by size. In addition, our approach provides a general framework for establishing new bounds. Received: August 11, 1998 / Accepted: February 1, 2001?Published online September 17, 2001     

超尺寸物品装箱问题及其算法

本文探讨一类新装箱问题-超尺寸物品装箱问题。针对实际解决该问题的两涉法，我们提出了一个评价效率更高的目标函数，证明了在此目标函数下两步法的渐近最坏比不小于2，并给出了渐近量坏比与拆分次数的关系。最后本文提出了一种不同于两步法的新在线算法MA，证明了在新目标函数下其渐近最坏比不超过7/4。     

Bin packing: On optimizing the number of pieces packed

We consider bin-packing variations related to the well-studied problem of maximizing the total number of pieces packed into a fixed set of bins. We show that, when the objective is to minimize the total number of pieces packed subject to the constraint that no unpacked piece will fit, no polynomial-time relative approximation algorithm exists (unless, of course,P=NP). That is, we prove that it isNP-hard to guarantee packing no more than a constant multiple of the optimal number of pieces, for any constant. This appears to be the first bin-packing problem for which this property has been demonstrated. Furthermore, this result also holds for the allied packing variant which seeks to minimize the maximum number of pieces packed in any single bin. We find the situation to be markedly better for the problem of maximizing the minimum number of pieces in any bin. If all bins possess the same capacity, we prove that the familiar SPF rule is an absolute approximation algorithm with additive constant 1, and can therefore be regarded as a best possible heuristic. For the more general and difficult case in which bin capacities may differ, it turns out that SPF fails to qualify as even a relative approximation algorithm. However, we devise an implementation of the well-known FFD heuristic, which we show to be a relative approximation algorithm, yielding a worst-case performance ratio of 1/2 with additive constant 0. Moreover, we prove that (unlessP=NP) no polynomial-time algorithm can guarantee a higher ratio without sacrificing the additive constant.This author's research is supported in part by the National Science Foundation under grants ECS-8403859 and MIP-8603879.     

Space defragmentation for packing problems

One of main difficulties of multi-dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed. This study proposes a defragmentation technique to combine the fragmented space into a continuous usable space, which potentially allows the packing of additional items. We illustrate the effectiveness of this technique using the two- and three-dimensional bin packing problem, where the aim is to load all given items (represented by rectangular boxes) into the minimum number of identical bins. Experimental results based on well-known 2D and 3D bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta-heuristic approaches for the problem. In conjunction with a bin shuffling strategy for incremental improvement, our resultant algorithm outperforms all leading meta-heuristic approaches based on the commonly used benchmark data by a significant margin.