Partially dynamic bin packing can be solved within 1 + ε in (amortized) polylogarithmic time

We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson. This problem is a generalization of the bin packing problem in which items may arrive and depart from the packing dynamically. The main result in this paper is a lower bound of 2.5 on the achievable competitive ratio, improving the best known 2.428 lower bound, and revealing that packing items of restricted form like unit fractions (i.e., of size 1/k for some integer k), for which a 2.4985-competitive algorithm is known, is indeed easier. We also investigate the resource augmentation version of the problem where the on-line algorithm can use bins of size b (>1) times that of the optimal off-line algorithm. An interesting result is that we prove b=2 is both necessary and sufficient for the on-line algorithm to match the performance of the optimal off-line algorithm, i.e., achieve 1-competitiveness. Further analysis gives a trade-off between the bin size multiplier 1<b≤2 and the achievable competitive ratio.

Cloud-based systems often face the problem of dispatching a stream of jobs to run on cloud servers in an online manner. Each job has a size that defines the resource demand for running the job. Each job is assigned to run on a cloud server upon its arrival and the job departs after it completes. The departure time of a job, however, is not known at the time of its arrival. Each cloud server has a fixed resource capacity and the total resource demand of all the jobs running on a server cannot exceed its capacity at all times. The objective of job dispatching is to minimize the total cost of the servers used, where the cost of renting each cloud server is proportional to its running hours by “pay-as-you-go” billing. The above job dispatching problem can be modeled as a variant of the dynamic bin packing (DBP) problem known as MinUsageTime DBP. In this paper, we study the competitiveness bounds of MinUsageTime DBP. We establish an improved lower bound on the competitive ratio of Any Fit family of packing algorithms, and a new upper bound of $\mu +3$ on the competitive ratio of the commonly used First Fit packing algorithm, where $\mu $ is the max/min job duration ratio. Our result significantly reduces the gap between the upper and lower bounds for the MinUsageTime DBP problem to a constant value independent of $\mu $ , and shows that First Fit packing is near optimal for MinUsageTime DBP.

The bin packing problem is one of the most fundamental optimization problems. Owing to its hardness as a combinatorial optimization problem class and its wide range of applications in different domains, different variations of the problem are emerged and many heuristics have been proposed for obtaining approximate solutions. In this paper, we solve a Multi-Level Bin Packing (MLBP) problem in the real make-to-order industry scenario. Existing solutions are not applicable to the problem due to: 1. the final packing may consist multiple levels of sub-packings; 2. the geometry shapes of objects as well as the packing constraints may be unknown. We design an automatic packing framework which extracts the packing knowledge from historical records to support packing without geometry shape and constraint information. Furthermore, we propose a dynamic programming approach to find the optimal solution for normal size problems; and a heuristic multi-level fuzzy-matching algorithm for large size problems. An inverted index is used to accelerate strategy search. The proposed auto packing framework has been deployed in Huawei Process & Engineering System to assist the packing engineers. It achieves a performance of accelerating the execution time of processing 5,000 packing orders to about $8$ minutes with an average successful packing rate as $80.54%$, which releases at least $30%$ workloads of packing workers.

Background: Logistics and transport, core of many business processes, are continuously optimized to improve efficiency and market competitiveness. The paper describes a modular coordination of vehicle routing and bin packing problems that enables independent instances of the problems to be joined together, with the aim that the vehicle routing solution satisfies all the constraints from real-world applications. Methods: The vehicle routing algorithm is based on an adaptive memory procedure that also incorporates a simple, one-dimensional bin packing problem. This preliminary packing solution is refined by a complex, three dimensional bin packing for each vehicle to identify the infeasible packages. The method iteratively adjusts virtual volumes until reaching near-optimal routes that respect bin-packing constraints. Results: The coordination enables independent applications of an adaptive memory procedure to vehicle routing and a genetic algorithm approach to bin packing while joining them in a computationally tractable way. Such a coordinated approach is applied to a frequently used public benchmark and proven to provide commensurate costs while significantly lowering algorithm complexity. Conclusions: The proposed method is further validated on a real industrial case study and provided additional savings of 14.48% in average daily distance traveled compared to the current industrial standard.

In this paper, we study the problem of how to dispatch the play requests to the cloud servers in a cloud gaming system. We show that the dispatching strategy of play requests may heavily affect the total service cost of the cloud gaming system. The play request dispatching problem can be considered as a variant of the dynamic bin packing problem. However, we show that the classical bin packing algorithms such as First Fit and Best Fit are not efficient in dispatching the play requests in cloud gaming due to the diurnal workload pattern of online games. To address this issue, we propose an efficient request dispatching algorithm that assigns the play requests according to the predicted ending times of the game sessions. Through extensive evaluations using real online gaming traces, we show that the proposed dispatching algorithm can significantly reduce the resource waste of the cloud servers and thus decrease the total service cost compared to the First Fit and Best Fit algorithms.

On data reduction for dynamic vector bin packing

In this paper, our objective is to develop a mathematical formulation of solving the Bin Packing Problem (BPP) with different sizes of box. It has become one of the most important mathematical applications throughout the time. In our study, Modified Branch and Bound Algorithm (MBBA) is developed to generate all the feasible packing patterns of given boxes to required containers for One-Dimensional BPP. Further development of algorithms was made to ascertain the locations of each box within the containers by using Cartesian coordinate system. Developed algorithms are coded and programmed in the Python programming environment to generate feasible packing patterns.

In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box $B$. The goal is to pack all the items in $B$ such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of $B$ and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any $\epsilon>0$, achieves an approximation of $T_{\infty}+\epsilon\approx1.69103+\epsilon$, where $T_{\infty}$ is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance $I$ and the strip packing solutions for the corresponding instance obtained from $I$ by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the $T_{\infty}+\epsilon$ approximation for 2D bin packing due to Caprara. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for 2D strip packing due to Kenyon and Rémila.

Dynamic bin packing of unit fractions items

This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of Ω( m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O ( p log 3 n ) (througout the paper the logarithms are based 2) for connectivity and bipartiteness. The worst-case time for one query is O (log n /log log n ). For the k -edge witness problem (“Does the removal of k given edges disconnect the graph?”) the expected time for p updates is O ( p log 3 n ) and the expected time for q queries is O ( qk log 3 n ). Given a graph with k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O ( pk log 3 n ). This implies an algorithm to maintain a 1 + ε-approximation of the minimum spanning tree in expected time O (( p log 3 n log U )/ε) for p updates, where the weights of the edges are between 1 and U .

The bin packing problem is a well known NP-hard problem that consists in placing a set of items of different sizes in a mimimum number of bins. An iterative algorithm for the one-dimensional offline bin packing problem is presented whose number of iterations needed to reach an unchanging partition when faced to a particular instance is bounded by the performance of the point Jacobi method on the problem taken as a matrix relaxation one. The algorithm was tested on a large widely used benchmark instances from the literature and behaved very successfully on a class considered as very difficult.

The study on one-dimensional bin packing problem may bring about many important applications such as multiprocessor scheduling, resource allocating, real-world planning and packing. Harmonic algorithms (including H K, RH, etc.) for bin packing have been famous for their linear-time and on-line properties for a long time. This paper profoundly analyzes the average-case performance of harmonic algorithms for pieces of i.i.d. sizes, provides the average-case performance ratio of H K under (0,d] (d ≤ 1) uniform distribution and the average-case performance ratio of RH under (0,1] uniform distribution. We also finished the discussion of the worst-case performance ratio of RH. Moreover, we propose a new improved version of RH that obtains better worst- and average-case performance ratios.

Using a Bin Packing Approach for Stowing Hazardous Containers into Containerships.- Dynamic Packing with Side Constraints for Datacenter Resource Management.- Packing Optimization of Free-Form Objects in Engineering Design.- A Modeling-Based Approach for Non-Standard Packing Problems.- CAST: A Successful Project in Support of the International Space Station Logistics.- Cutting and Packing Problems with Placement Constraints.- A Container Loading Problem MILP-based Heuristics Solved by CPLEX:An Experimental Analysis.- Automatic Design of Optimal LED Street Lights.- Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting.- Exploiting Packing Components in General-Purpose Integer Programming Solvers.- Robust Designs for Circle Coverings of a Square.- Batching-based Approaches for Optimized Packing of Jobs in the Spatial Scheduling Problem.- Optimized Object Packings Using Quasi-Phi-Functions.- Graph Coloring Models and Metaheuristics for Packing Applications.

As one of the core technology of ASON (automatically switched optical network), the dynamic routing and wavelength assignment algorithm is the key for ASON to reflecting its intelligence. In this paper, a new distributed heuristic algorithm based ant colony optimization for dynamic RWA (routing and wavelength assignment) that is adapted to the environment of ASON networks was put forward. The route selecting and the wavelength assignment process were combined with as whole using multilayer-graph model. The algorithm implements in distributed way using local information stored in a pheromone matrix and will be refreshed on the way of searching path. Compare with shortest-path+first-fit algorithms, it can reduce communication overhead cost of networks and the simulation computing showed lower blocking probability and more balanced wavelength resource assignment.

This work aims to study and explore the use of Gene Expression Programming (GEP) in solving on-line Bin-Packing problem. The main idea is to show how GEP can automatically find acceptable heuristic rules to solve the problem efficiently and economically. One dimensional Bin-Packing problem is considered in the course of this work with the constraint of minimizing the number of bins filled with the given pieces. Experimental Data includes instances of benchmark test data taken from Falkenauer (1996) for One-dimensional Bin-Packing Problems. Results show that GEP can be used as a very powerful and flexible tool for finding interesting compact rules suited for the problem. The impact of functions is also investigated to show how they can affect and influence the success of rates when they appear in rules. High success rates are gained with smaller population size and fewer generations compared to a previous work performed using Genetic Programming.