Polynomial time approximation schemes for class-constrained packing problems

Approximation and online algorithms for multidimensional bin packing: A survey

We consider a natural generalization of the knapsack problem and the multiple knapsack problem, which has two phases of packing decisions. In this problem, we have a set of items, several small knapsacks called boxes, and a large knapsack called container. Each item has a size and profit, each box has a size and the container has a capacity. The first phase is to select some items to pack into the boxes, and the second phase is to select the boxes (each includes some packed items) to pack into the container. The total profit of the problem is determined by the items that are selected and packed into the container within some packed box, and the objective is to maximize the total profit. This problem is motivated by various practical applications, e.g., in logistics. It is a generalization of the multiple knapsack problem, and hence is strongly NP-hard. We mainly propose three approximation algorithms for it. Particularly, the first one is a \(\frac{1}{4}\)-approximation algorithm based on its linear programming relaxation; the second one is based on applying the algorithms for the multiple knapsack problem and the knapsack problem, and has an approximation ratio \(\frac{1}{3} - \epsilon \) for any small enough \(\epsilon >0\). We finally provide a polynomial time approximation scheme for this problem.

A faster FPTAS for knapsack problem with cardinality constraint

We consider two variants of the classical bin packing problem in which items may be fragmented. This can potentially reduce the total number of bins needed for packing the instance. However, since fragmentation incurs overhead, we attempt to avoid it as much as possible. In bin packing with size increasing fragmentation (BP-SIF), fragmenting an item increases the input size (due to a header/footer of fixed size that is added to each fragment). In bin packing with size preserving fragmentation (BP-SPF), there is a bound on the total number of fragmented items. These two variants of bin packing capture many practical scenarios, including message transmission in community TV networks, VLSI circuit design and preemptive scheduling on parallel machines with setup times/setup costs.While both BP-SPF and BP-SIF do not belong to the class of problems that admit a polynomial time approximation scheme (PTAS), we show in this paper that both problems admit a dual PTAS and an asymptotic PTAS. We also develop for each of the problems a dual asymptotic fully polynomial time approximation scheme (AFPTAS). The AFPTASs are based on a non-trivial application of a fast combinatorial FPTAS for packing linear programs with negative entries, proposed recently by Garg and Khandekar [5].KeywordsApproximation SchemeParallel MachinePolynomial Time Approximation SchemeLarge ItemSmall 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.

Profitable tour problem (PTP) belongs to the class of vehicle routing problem (VRP) with profits seeking to maximize the difference between the total collected profit and the total cost incurred. Traditionally, PTP involves single vehicle. In this paper, we consider PTP with multiple vehicles. Unlike the classical VRP that seeks to serve all customers, PTP involves the strategic-level customer selection so as to maximize the total collected profit and the operational-level route optimization to minimize the total cost incurred. Therefore, PTP is essentially the knapsack problem at the strategic level with VRP at the operational level. That means the evolutionary bi-level programming would be a suitable choice of methodology for solving the NP-hard PTP. Employing some evolutionary method to solve the bi-level program naively would undoubtedly be prohibitively expensive. We thus present in this paper the notion of knowledge adoption to approximate the initial solution to the lower-level optimization problem for a given trial solution of the upper-level decision variables. One may consider the knowledge adoption as a special case of knowledge transfer in which the transfer takes place within the same problem domain. Refining the approximate initial solution with local search forces it to quickly converge to some locally optimal solution. The better the estimation of the initial solution, the closer the local optimum will be to the global one. PTP finds its important application in the fields of transportation and logistics. In addressing last-mile problem using auction at the urban consolidation center (UCC), PTP plays a significant role in the winner determination problem (WDP) that follows. Empirical study demonstrates the efficacy of the proposed approach in solving the PTP-based WDP, yielding significantly higher profit, utilization, and service level compared to when the UCC adopts the conventional WDP based on multiple knapsack problem (i.e. the MKP-based WDP).

The multiple knapsack problem (MKP) is a classical combinatorial optimization problem. A recent algorithm for some classes of the MKP is bin-completion, a bin-oriented, branch-and-bound algorithm. In this paper, we propose path-symmetry and path-dominance criteria for pruning nodes in the MKP branch-and-bound search space. In addition, we integrate the “bound-and-bound” upper bound validation technique used in previous MKP solvers. We show experimentally that our new MKP solver, which successfully integrates dominance based pruning, symmetry breaking, and bound-and-bound, significantly outperforms previous solvers on some classes of hard problem instances.

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.

PurposeIn the literature, there are many methods that can be helpful in strategic management of universities. Some of them are related to the aspect of sustainability, in terms of balancing the level of fulfillment of different, often conflicting objectives, which must be considered when building strategies. These methods include product/service portfolio ones. However, their use is often intuitive and detached from the quantitative aspects of management. The purpose of this paper is to present a proposal of the modification of the portfolio methods through the use of one of discrete optimization problems, namely, the multiple knapsack problem. The proposal is applied to a selected university. The results are presented and discussed.Design/methodology/approachThe research methodology consists in a conceptual work: combining non-quantitative portfolio methods used in strategic management and the quantitative multiple knapsack problem. The analogy is established between a market sector (capacity) and a knapsack (capacity), a university department value and an object value, the university improvement budget and a knapsack, an object cost and an improvement measure cost. Then, the case study is used to conduct an initial validation of the proposed approach.FindingsA quantitative model for strategic management of university as a whole or university departments is proposed. It allows to plan and control the application of improvement measures, allowing the university units to take on a better position in the educational market. It has been initially applied to a small university.Research limitations/implicationsThe model requires much more real-world case studies. Also, it will usually be difficult to establish the cost of running individual university units as well as the cost of corrective measures. The capacity of knapsacks – here of market sectors – will also be difficult to calculate. The method to be used here is activity-based costing, but it will not solve all the problems immediately, as its practical application is difficult.Practical implicationsThe proposed model will allow to plan and evaluate strategic changes of university as a whole and its units’ position using quantitative values and to consider various strategic scenarios.Social implicationsIn order to establish the data necessary to construct the model, various stakeholders will have to cooperate (the university promotion department, the accounting representatives, the student and industry representatives etc.), probably for the first time. Such cooperation will improve the university position even if the model itself will not be able to be applied immediately.Originality/valueThe link between portfolio methods and a quantitative optimization method for university management purposes has been established in the paper for the first time in the literature.

The multiple knapsack problem with grouped items aims to maximize rewards by assigning groups of items among multiple knapsacks, without exceeding knapsack capacities. Either all items in a group are assigned or none at all. We study the bi-criteria variation of the problem, where capacities can be exceeded and the second objective is to minimize the maximum exceeded knapsack capacity. We propose approximation algorithms that run in pseudo-polynomial time and guarantee that rewards are not less than the optimal solution of the capacity-feasible problem, with a bound on exceeded knapsack capacities. The algorithms have different approximation factors, where no knapsack capacity is exceeded by more than 2, 1, and \(1/2\) times the maximum knapsack capacity. The approximation guarantee can be improved to \(1/3\) when all knapsack capacities are equal. We also prove that for certain cases, solutions obtained by the approximation algorithms are always optimal—they never exceed knapsack capacities. To obtain capacity-feasible solutions, we propose a binary-search heuristic combined with the approximation algorithms. We test the performance of the algorithms and heuristics in an extensive set of experiments on randomly generated instances and show they are efficient and effective, i.e., they run reasonably fast and generate good quality solutions.

Understanding the interactions between different combinatorial optimisation problems in real-world applications is a challenging task. Recently, the traveling thief problem (TTP), as a combination of the classical traveling salesperson problem and the knapsack problem, has been introduced to study these interactions in a systematic way. We investigate the underlying non-linear packing while traveling (PWT) problem of the TTP where items have to be selected along a fixed route. We give an exact dynamic programming approach for this problem and a fully polynomial time approximation scheme (FPTAS) when maximising the benefit that can be gained over the baseline travel cost. Our experimental investigations show that our new approaches outperform current state-of-the-art approaches on a wide range of benchmark instances.

We present the first polynomial-time approximation scheme (PTAS), i.e., (1+e)-approximation algorithm for any constant e> 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn (loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing light-weight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems. Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners for planar TSP by Klein [SIAM J. Computing 2008], subset TSP by Klein [STOC 2006], Steiner tree by Borradaile, Klein, and Mathieu [ACM Trans. Algorithms 2009], and Steiner forest by Bateni, Hajiaghayi, and Marx [J. ACM 2011] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution. Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2+)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge.

The spectrum allocation for cognitive radio networks (CRNs) has received considerable studies under the assumption that the bandwidth of spectrum holes is static. However, in practice, the bandwidth of spectrum holes is time-varied due to primary user/secondary user (PU/SU) activity and mobility, which result in non-determinacy. This paper studies the spectrum allocation for CRNs with non-deterministic bandwidth of spectrum holes. We present a novel probability density function (PDF) model through order statistic to describe the non-deterministic bandwidth of spectrum holes and provide a bound to approximate it. After that, a statistical spectrum allocation model based on stochastic multiple knapsack problem (MKP) is established for spectrum allocation with non-deterministic bandwidth of spectrum holes. To reduce the computational complexity, we transform this stochastic programming probleminto a constant MKP though exploiting the properties of cumulative distribution function (CDF), which can be solved via MTHG algorithm by using auxiliary variable. Simulation results illustrate that the proposed statistical spectrum allocation algorithm can achieve better performances compared to the existing algorithms when the bandwidth of spectrum holes istime-varied.

We propose a new evolutionary algorithm for the multiple knapsack problem (MKP) which uses multiple representations. Previous, successful approaches for the MKP have included a weight-coded, order-based representation, as well as a grouping representation enhanced by a dominance condition to restrict search. We propose a representation-switching genetic algorithm which periodically transforms the representation of individuals between these two representations. We show that this new hybrid algorithm outperforms the previous approaches.

An extension of the classic Knapsack Problem (KP) is the Multiple Knapsack Problem with Setups (MKPS). The MKPS has been used to model business applications that have previously been solved using approximate solution methods and implemented in industrial settings. In the operations research (OR) literature, there are a number of sophisticated solution approaches that, based on 360 MKPS instances typically used to test MKPS solution method performance, generate very good solutions. However, codes for these customized solution methods are typically not available for use by OR practitioners. In this paper, we use these same MKPS instances to demonstrate how well simple sequential increasing tolerance (SSIT) strategies combined with general-purpose integer programming software (Gurobi) can efficiently solve these MKPS instances. More importantly, it is shown both empirically and statistically that these simple approaches are as good as the best specialized methods in the literature. Hence, we present a methodology that is both easy for the OR practitioner to use and which gives results competitive with the best specialized MKPS solution methods in the literature without the need to generate algorithm-specific code. Finally, our approach is the only one that guarantees how close our solutions are to the optimums.

In this work, we propose modified versions of shuffled frog leaping algorithm (SFLA) to solve multiple knapsack problems (MKP). The proposed algorithm includes two important operations: repair operator and genetic mutation with a small probability. The former is utilizing the pseudo-utility to repair infeasible solutions, and the later can effectively prevent the algorithm from trapping into the local optimal solution. Computational experiments with a large set of instances show that the proposed algorithm can be an efficient alternative for solving 0/1 multidimensional knapsack problem.