Approximation Schemes for Packing with Item Fragmentation

We consider the train delivery problem which is a generalization of the bin packing problem and is equivalent to a one dimensional version of the vehicle routing problem with unsplittable demands. The problem is also equivalent to the problem of minimizing the makespan on a single batch machine with non-identical job sizes. The train delivery problem is strongly NP-hard and does not admit an approximation ratio better than 3/2. We design the first approximation schemes for the general problem. We give an asymptotic polynomial time approximation scheme, under a notion of asymptotic that makes sense even though scaling can cause the cost of the optimal solution of any instance to be arbitrarily large. Alternatively, we give a polynomial time approximation scheme for the case where W, an input parameter that corresponds to the bin size or the vehicle capacity, is polynomial in the number of items or demands.

Square bin packing, where square items are to be packed into a minimum number of unit size squares, and its multidimensional generalization, cube packing, where $d$-dimensional cubic items are to be packed into a minimum number of unit size cubes of the same dimension, are well-studied generalizations of the classical one-dimensional bin packing problem. These problems are known to have asymptotic polynomial time approximation schemes (APTAS). In this paper we design robust approximation schemes for these problems. At each step, a robust algorithm receives a single input item to be added to the packing. The new item must be packed and the packing can be modified, but the total volume of items which may migrate between bins, or change their positions inside bins, must be bounded by a constant factor times the volume of the new item. That is, the solution is created by slightly adjusting the solution upon the arrival of each new item. Previous approximation schemes partition items into three types according...

The 2-dimensional bin packing problem (2BP) is a generalization of the classical Bin Packing problem and is defined as follows: Given a collection of rectangles specified by their width and height, pack these into the minimum number of square bins of unit size. We study the case of 'orthogonal packing without rotations', where rectangles cannot be rotated and must be packed parallel to the edges of a bin. Often in practical cases of 2BP problems there are additional constraints on how complicated the packing patterns in a bin can be. A well-studied and frequently used constraint is that every rectangle in the packing must be obtainable by recursively applying a sequence of edge-to-edge cuts parallel to the edges of the bin. Such cuts are known as guillotine cuts. Our main result is that the guillotine 2BP problem admits an asymptotic polynomial time approximation scheme. This is in sharp contrast with the fact that the general 2BP problem is APX-Hard. En route to our main result, we show a structural theorem about approximating general guillotine packings by simpler packings, which could be of independent interest.

Approximation and online algorithms for multidimensional bin packing: A survey

We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colours) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can accommodate. In the class-constrained multiple knapsack (CCMK) problem, our goal is to maximize the total value of packed items, whereas in the class-constrained bin-packing (CCBP), we seek to minimize the number of (identical) bins, needed for packing all the items. We give a polynomial time approximation scheme (PTAS) for CCMK and a dual PTAS for CCBP. We also show that the 0–1 class-constrained knapsack admits a fully polynomial time approximation scheme, even when the number of distinct colours of items depends on the input size. Finally, we introduce the generalized class-constrained packing problem (GCCP), where each item may have more than one colour. We show that GCCP is APX-hard, already for the case of a single knapsack, where all items have the same size and the same value. Our optimization problems have several important applications, including storage management for multimedia systems, production planning, and multiprocessor scheduling. Copyright © 2001 John Wiley & Sons, Ltd.

Multiprocessor platforms have been widely adopted in both embedded and server systems. In addition to the performance improvement, multiprocessor systems could have the flexibility in tolerating processor failures via task replication. This paper considers the replication of periodic hard real-time tasks in identical multiprocessor environments. Each task is replicated on K distinct processors, where K is a user-determined integer for fault tolerance to improve system reliability. When the objective is to minimize the maximum utilization in a system with a specified number of processors, we present a greedy algorithm with a 2-approximation ratio, and a polynomial-time approximation scheme is developed. For the minimization of the number of processors required to derive feasible schedules with task replication, we develop greedy algorithms with a 2-approximation ratio and an asymptotic polynomial-time approximation scheme

We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund (1997)) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well-known classical graph problems and include the minimal spanning tree problem, a number of elementary machine scheduling problems, the bin-packing problem, and the travelling salesman problem (TSP). We conclude that for all these generalized problems the existence of a polynomial time approximation scheme (PTAS) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality 2.

A faster FPTAS for knapsack problem with cardinality constraint

We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machines so as to minimize the makespan. In our problem, a job has its original processing time, which could be reduced by utilizing one of the resources. As resources are different, the amount of the time reduced for each job is different depending on the resource it uses. Once a resource is being used by one job, it can not be used simultaneously by any other job until this job is finished, hence the scheduler should take into account the job-to-machine assignment together with the resource-to-job assignment. We observe that, the classical unrelated machine scheduling problem is actually a special case of our problem when m=c, i.e., the number of resources equals the number of machines. Extending the techniques for the unrelated machine scheduling, we give a 2-approximation algorithm when both m and c are part of the input. We then consider two special cases for the problem, with m or c being a constant, and derive PTASes (Polynomial Time Approximation Schemes) respectively. We also establish the relationship between the two parameters m and c, through which we are able to transform the PTAS for the case when m is constant to the case when c is a constant. The relationship between the two parameters reveals the structure within the problem, and may be of independent interest.

Motivated by urban freight transportation practices in China, we study an optimal truck loading problem in which a fixed cost and an additional cost that depends on the number of unloading points are associated with each truck used. The truck loading problem is modeled as a one-dimensional bin packing problem, where the cost of each bin is a convex fixed-plus-linear function of the number of items in the bin. The objective is to minimize the total cost of bins used. We develop an asymptotic polynomial time approximation scheme and an efficient approximation algorithm with an asymptotic worst-case performance ratio of 1.5 to tackle the problem. Our computational experiments indicate that the approximation algorithm performs well for certain order patterns, and also reveal several important insights on using the freight charge scheme.

#NFA refers to the problem of counting the words of length n accepted by a non-deterministic finite automaton. #NFA is #P-hard, and although fully-polynomial-time randomized approximation schemes (FPRAS) exist, they are all impractical. The first FPRAS for #NFA had a running time of Õ(n 17 m 17 ε -14 łog(δ -1 )), where m is the number of states in the automaton, δ ∈ (0,1] is the confidence parameter, and ε > 0 is the tolerance parameter (typically smaller than 1). The current best FPRAS achieved a significant improvement in the time complexity relative to the first FPRAS and obtained FPRAS with time complexity Õ((n 10 m 2 + n 6 m 3 )ε -4 łog 2 (δ -1 )). The complexity of the improved FPRAS is still too intimidating to attempt any practical implementation. In this paper, we pursue the quest for practical FPRAS for #NFA by presenting a new algorithm with a time complexity of O(n 2 m 3 łog(nm)ε -2 łog(δ -1 )). Observe that evaluating whether a word of length n is accepted by an NFA has a time complexity of O(nm 2 ). Therefore, our proposed FPRAS achieves sub-quadratic complexity with respect to membership checks.

Approximation schemes for parallel machine scheduling with availability constraints

In this paper we deal with the d-dimensional vector packing problem, which is a generalization of the classical bin packing problem in which each item has d distinct weights and each bin has d corresponding capacities. We address the case in which the vectors of weights associated with the items are totally ordered, i.e., given any two weight vectors a i, a j, either a i is componentwise not smaller than a j or a j is componentwise not smaller than a i, and construct an asymptotic polynomial-time approximation scheme for this case. As a corollary, we also obtain such a scheme for the bin packing problem with cardinality constraint, whose existence was an open question to the best of our knowledge.We also extend the result to instances with constant Dilworth number, i.e. instances where the set of items can be partitioned into a constant number of totally ordered subsets. We use ideas from classical and recent approximation schemes for related problems, as well as a nontrivial procedure to round an LP solution associated with the packing of the small items.KeywordsApproximation SchemePacking ProblemOptimal AssignmentCardinality ConstraintLarge ItemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the Ordered Open End Bin Packing problem. Items of sizes in (0, 1] are presented one by one, to be assigned to bins in this order. An item can be assigned to any bin for which the current total size is strictly below 1. This means also that the bin can be overloaded by its last packed item. We improve lower and upper bounds on the asymptotic competitive ratio in the online case. Specifically, we design the first algorithm whose asymptotic competitive ratio is strictly below 2, and its value is close to the lower bound. This is in contrast to the best possible absolute competitive ratio, which is equal to 2. We also study the offline problem where the sequence of items is known in advance, while items are still assigned to bins based on their order in the sequence. For this scenario, we design an asymptotic polynomial time approximation scheme.

PurposeThe purpose of this paper is to find the best approximation algorithm for solving the more general case of single‐supplier multi‐retailer capacitated economic lot‐sizing (SM‐CELS) problem in deterministic inventory theory, which is the non‐deterministic polynomial (NP)‐hard problem.Design/methodology/approachSince few theoretical results have been published on polynomial time approximation algorithms for SM‐CELS problems, this paper develops a fully polynomial time approximation scheme (FPTAS) for the problem with monotone production and holding‐backlogging cost functions. First the optimal solution of a rounded problem is presented as the approximate solution and its straightforward dynamic‐programming (DP) algorithm. Then the straightforward DP algorithm is converted into an FPTAS by exploiting combinatorial properties of the recursive function.FindingsAn FPTAS is designed for the SM‐CELS problem with monotone cost functions, which is the strongest polynomial time approximation result.Research limitations/implicationsThe main limitation is that the supplier only manufactures without holding any products when the model is applied.Practical implicationsThe paper presents the best result for the SM‐CELS problem in deterministic inventory theory.Originality/valueThe LP‐rounding technique, an effective approach to design approximation algorithms for NP‐hard problems, is successfully applied to the SM‐CELS problem in this paper.