Knapsack Problem Research Articles

Improved approximation for two-dimensional vector multiple knapsack

We study the uniform 2-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional weight vector and a positive profit, along with m 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.Our main result is a (1−ln⁡22−ε)-approximation algorithm for 2VMK, for every fixed ε>0, thus improving the best known ratio of (1−1e−ε) which follows as a special case from a result of Fleischer et al. (2011) [6].Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) [15], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to ≈m⋅ln⁡2≈0.693⋅m of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.

Cooperative coati optimization algorithm with transfer functions for feature selection and knapsack problems

Cooperative coati optimization algorithm with transfer functions for feature selection and knapsack problems

Energy-efficient scheduling in an identical parallel machine environment with peak power consumption and deadline constraints

Energy-efficient scheduling is an essential means of achieving sustainability in manufacturing systems. This paper defines and addresses the problem of scheduling n jobs on m identical parallel machines in which peak power consumption and deadline constraints exist simultaneously. The objective is to maximize the total value of the selected jobs. We show that this problem is equivalent to a special case of the rectangular knapsack problem, based on which four properties are observed. To solve the problem, an effective mixed integer linear programming model is proposed based on the properties, and it is much more efficient compared to the performance of modeling methods inspired by other works. Furthermore, three effective decoding methods are proposed and embedded into genetic algorithms (GAs). Comparisons with the exact algorithm (i.e., the proposed model) show that our GAs can lead to good-quality solutions within one second for small instances. Meanwhile, experimental results for large instances indicate that the proposed GAs can obtain near-optimal or satisfactory solutions. Finally, results also show that the proposed GA-DD significantly outperforms the existing matheuristic algorithm.

A novel discrete differential evolution algorithm combining transfer function with modulo operation for solving the multiple knapsack problem

In this paper, an efficient method for solving multiple knapsack problem (MKP) using discrete differential evolution is proposed. Firstly, an integer programming model of MKP suitable for discrete evolutionary algorithm is established. Secondly, a new method for discretizing continuous evolutionary algorithm is proposed based on the combination of transfer function and modulo operation. Therefrom, a new discrete differential evolution algorithm (named TMDDE) is proposed. Thirdly, the algorithm GROA is proposed to eliminate infeasible solutions of MKP. On this basis, a new method for solving MKP using TMDDE is proposed. Finally, the performance of TMDDE using S-shaped, U-shaped, V-shaped, and Taper-shaped transfer functions combined with modulo operation is compared, respectively. It is pointed out that T3-TMDDE which used Taper-shaped transfer function T3 is the best. The comparison results of solving 30 MKP instances show that the performance of T3-TMDDE is better than five advanced evolutionary algorithms. It not only indicates that TMDDE is more competitive for solving MKP, but also demonstrates that the proposed discretization method is an effective method.

Time efficiency and match optimization in scheduling Pencak Silat matches: a case study of the Sleman Regency Student Pencak Silat Championship in 2023

The issue of this research is that many pencak silat matches in Indonesia still use manual calculations in several dis-tricts/regencys, then the calculation data is inputted into the system. This will take a lot of time, so the pencak silat match schedule spends a lot of time. The data used is the number of parties at the Sleman Regency Student Pencak Silat Championship in 2023. This research aims to find calculations to maximize the number of parties/matches/match numbers per day in a martial arts match. The method that can be used is the Preemptive Multiple Bounded Knapsack Problem (PMBKP). PMBKP involves distributing the quanti-ty of different types of items to a group of knapsacks with their respective priorities and capacity limits. The PMBKP results obtained in the 2023 Sleman District Pencak Silat Championship problem using the Branch and Bound algorithm are 175 parties on the first day, 133 parties on the second day, 128 parties on the third day, 127 parties on the fourth day, and 72 parties on the fifth day. These results can be said to be effective if applied to pencak silat matches that aim to nurture achievements, such as introductions and district/regency level matches. Keywords: Pencak Silat, Competition, Sports Schedule

Energy‐efficient adaptive dependent task scheduling in cooperative vehicle‐infrastructure system

AbstractIn the cooperative vehicle‐infrastructure system (CVIS), due to its computation limitation, vehicles are difficult to handle computing‐intensive delay‐sensitive tasks, so offload tasks to roadside unit (RSU) become popular. Due to the complexity of vehicles’ tasks and tasks generated by different vehicles have different delay constraints, minimize energy consumption of RSUs under task dependence and delay constraints is challenging. This paper defines the task priority queuing criterion for the task priority division problem, proposes a task scheduling strategy for energy‐packet queue length tradeoff (TSET) in CVIS under RSUs distributed task scheduling problem and establishes the vehicle speed state model, task model, data queue model, task computing model and energy consumption model. After Lyapunov optimization theory transformed the optimization model, a knapsack problem was described. The simulation results verify that TSET reduces the average energy consumption of roadside units and ensures the stability of the data queue under task dependence and deadline conditions.

Parallel implementation of an exact two-phase method for the biobjective knapsack problem

Parallel implementation of an exact two-phase method for the biobjective knapsack problem

Finding and exploring promising search space for The 0–1 Multidimensional Knapsack Problem

The 0–1, Multidimensional Knapsack Problem (MKP) is a classical NP-hard combinatorial optimization problem with many engineering applications. In this paper, we propose a novel algorithm combining evolutionary computation with the exact algorithm to solve the 0–1 MKP. It maintains a set of solutions and utilizes the information from the population to extract good partial assignments. To find high-quality solutions, an exact algorithm is applied to explore the promising search space specified by the good partial assignments. The new solutions are used to update the population. Thus, the good partial assignments evolve towards a better direction with the improvement of the population. Extensive experimentation with commonly used benchmark sets shows that our algorithm outperforms the state-of-the-art heuristic algorithms, TPTEA and DQPSO, as well as the commercial solver CPlex. It finds better solutions than the existing algorithms and provides new lower bounds for 10 large and hard instances.

Exploring Heuristic Algorithms for the Knapsack Problem: A Comparative Analysis of Program Complexity and Computational Efficiency

Exploring Heuristic Algorithms for the Knapsack Problem: A Comparative Analysis of Program Complexity and Computational Efficiency

Effective incentive based demand response with voltage support capability via reinforcement learning based multi-agent framework

With the ongoing advancements in power grids, it is crucial to maintain a balance between energy generation and usage to ensure the stability of power systems. Any disparity between the supply and demand of energy can lead to increased expenses for utility companies and consumers, which might potentially result in a widespread failure of the grid. To tackle this problem, this study offers the implementation of the Voltage Capability Incentive-Based Demand Response (VC-IBDR) system, which utilizes a multi-agent framework. The main goal is to reduce power usage by providing financial incentives, while also ensuring that voltage standards are satisfied at each node. In this method, the aggregator agent coordinates demand response operations by interacting with household agents that are equipped with home energy management systems. Every household utilizes a disjunctively constrained knapsack problem optimization method to effectively organize the consumption of home appliances, considering customer preferences and dissatisfaction. During demand response events, the aggregator agent works together with household agents to optimize appliance schedules, ensuring that voltage levels at each node remain within acceptable levels. The simulation is performed to assess the efficacy of VC-IBDR on a feeder that provides service to 25 residential households. The findings indicate substantial decreases in power usage, with an average decline in demand of 4.20% at the feeder level and a 10.41% decrease in peak power consumption throughout the day. Furthermore, the voltage values at each node stay consistently steady within the range of 0.96 to 1.0 per unit throughout the day.

New classes of Facets for complementarity knapsack problems

The complementarity knapsack problem (CKP) is a knapsack problem with real-valued variables and complementarity conditions between pairs of variables. We extend the polyhedral studies of de Farias et al. for CKP, by proposing three new families of cutting-planes that are obtained from a combinatorial concept known as pack. Sufficient conditions for these inequalities to be facet-defining, based on the new concept of a maximal switching pack, are also provided. Moreover, we answer positively a conjecture by de Farias et al. about the separation complexity of the inequalities introduced in their work, and propose efficient separation algorithms for our newly defined cutting-planes.

An Improved Unbounded-DP Algorithm for the Unbounded Knapsack Problem with Bounded Coefficients

Benchmark instances for the unbounded knapsack problem are typically generated according to specific criteria within a given constant range R, and these instances can be referred to as the unbounded knapsack problem with bounded coefficients (UKPB). In order to increase the difficulty of solving these instances, the knapsack capacity C is usually set to a very large value. While current efficient algorithms primarily center on the Fast Fourier Transform (FFT) and (min,+)-convolution method, there is a simpler method worth considering. In this paper, based on the basic Unbounded-DP algorithm, we utilize a recent branch and bound (B&B) result and basic theory of linear Diophantine equation, and propose an improved Unbounded-DP algorithm with time complexity of O(R4) and space complexity of O(R3). Additionally, the algorithm can also solve the All-capacities unbounded knapsack problem within the complexity O(R4+C). In particular, the proof techniques required by the algorithm are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.

An Improved Elite Correlation Selection Using the Correlation Tournament Selection in Genetic Algorithm

In this paper, we propose an improved Elite Correlation Selection using the Correlation Tournament Selection in Genetic Algorithm. The new algorithm is applied to a Variable Knapsack Problem and shows a better performance. Keywords: Correlation Tournament Selection, Improved Elite Correlation Selection, Genetic Algorithm, Variable Knapsack Problem

Binary metaheuristic algorithms for 0–1 knapsack problems: Performance analysis, hybrid variants, and real-world application

This paper examines the performance of three binary metaheuristic algorithms when applied to two distinct knapsack problems (0–1 knapsack problems (KP01) and multidimensional knapsack problems (MKP)). These binary algorithms are based on the classical mantis search algorithm (MSA), the classical quadratic interpolation optimization (QIO) method, and the well-known differential evolution (DE). Because these algorithms were designed for continuous optimization problems, they could not be used directly to solve binary knapsack problems. As a result, the V-shaped and S-shaped transfer functions are used to propose binary variants of these algorithms, such as binary differential evolution (BDE), binary quadratic interpolation optimization (BQIO), and binary mantis search algorithm (BMSA). These binary variants are evaluated using various high-dimensional KP01 examples and compared to several classical metaheuristic techniques to determine their efficacy. To enhance the performance of those binary algorithms, they are combined with repair operator 2 (RO2) to offer better hybrid variants, namely HMSA, HQIO, and HDE. Those hybrid algorithms are evaluated using several medium- and large-scale KP01 and MKP instances, as well as compared to other hybrid algorithms, to demonstrate their effectiveness. This comparison is conducted using three performance metrics: average fitness value, Friedman mean rank, and computational cost. The experimental findings demonstrate that HQIO is a strong alternative for solving KP01 and MKP. In addition, the proposed algorithms are applied to the Merkle-Hellman Knapsack Cryptosystem and the resource allocation problem in adaptive multimedia systems (AMS) to illustrate their effectiveness when applied to optimize those real applications. The experimental findings illustrate that the proposed HQIO is a strong alternative for handling various knapsack-based applications.

Online Unit Profit Knapsack with Predictions

A variant of the online knapsack problem is considered in the setting of predictions. In Unit Profit Knapsack, the items have unit profit, i.e., the goal is to pack as many items as possible. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. The prediction available to the online algorithm is the average size of those smallest items that fit in the knapsack. For the prediction error in this hard online problem, we use the ratio r=aa^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r=\\frac{a}{\\hat{a}}$$\\end{document} where a is the actual value for this average size and a^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{a}$$\\end{document} is the prediction. 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More generally, the algorithm has a competitive ratio of e-1er\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{e-1}{e}r$$\\end{document}, if r≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\le 1$$\\end{document}, and e-rer\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{e-r}{e}r$$\\end{document}, if 1≤r<e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le r < e$$\\end{document}. 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To obtain a positive competitive ratio for all r, we adjust the algorithm, resulting in a competitive ratio of 12r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2r}$$\\end{document} for r≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 1$$\\end{document} and r2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{r}{2}$$\\end{document} for r≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 1$$\\end{document}. We show that improving the result for any r<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r< 1$$\\end{document} leads to a worse result for some r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}.

Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack

In “Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack,” Jiang, Ma, and Zhang analyze the multiunit prophet inequalities and online knapsack problems. They formulate the problems as the online contention resolution scheme problems. Then, they develop linear programming formulations to represent the online contention resolution scheme problems. They characterize the optimal solution of the linear programming and derive the corresponding optimal policies. For the multiunit prophet inequalities, they are able to obtain the tight ratios, which improved over the previous ones. For the online stochastic knapsack problem, their ratio is also optimal and the best up to date. They finally generalize their results to the multiresource setting and expand the applicability of their approach.

An adaptive evolutionary search-based method for efficiently tackling the set-union knapsack problem

This paper tackles the NP-hard set-union knapsack problem using a novel adaptive evolutionary algorithm. Such a problem has applications in database query optimization, network design, and project portfolio management. The method starts by creating an archive set with elite and diversified solutions, employing a customized procedure based on item scores and Hamming max-min distance for diversity. Various operators are added to enhance solution quality in the elite set and refine the search process. To counteract premature convergence, combination and subset reference update methods are used. The fusion strategy serves as an exploration mechanism, generating new solutions and preventing premature convergence. Finally, the effectiveness and performance of the proposed method is evaluated on a set of benchmark instances taken from existing literature and by conducting a thorough statistical analysis of all achieved results. All obtained results are systematically compared against those achieved by state-of-the-art methods, indicating a substantial improvement of 50% for the large-sized instances, outperforming those published in the literature.

PEGA: A Privacy-Preserving Genetic Algorithm for Combinatorial Optimization.

Evolutionary algorithms (EAs), such as the genetic algorithm (GA), offer an elegant way to handle combinatorial optimization problems (COPs). However, limited by expertise and resources, most users lack the capability to implement EAs for solving COPs. An intuitive and promising solution is to outsource evolutionary operations to a cloud server, however, it poses privacy concerns. To this end, this article proposes a novel computing paradigm called evolutionary computation as a service (ECaaS), where a cloud server renders evolutionary computation services for users while ensuring their privacy. Following the concept of ECaaS, this article presents privacy-preserving genetic algorithm (PEGA), a privacy-preserving GA designed specifically for COPs. PEGA enables users, regardless of their domain expertise or resource availability, to outsource COPs to the cloud server that holds a competitive GA and approximates the optimal solution while safeguarding privacy. Notably, PEGA features the following characteristics. First, PEGA empowers users without domain expertise or sufficient resources to solve COPs effectively. Second, PEGA protects the privacy of users by preventing the leakage of optimization problem details. Third, PEGA performs comparably to the conventional GA when approximating the optimal solution. To realize its functionality, we implement PEGA falling in a twin-server architecture and evaluate it on two widely known COPs: 1) the traveling Salesman problem (TSP) and 2) the 0/1 knapsack problem (KP). Particularly, we utilize encryption cryptography to protect users' privacy and carefully design a suite of secure computing protocols to support evolutionary operators of GA on encrypted chromosomes. Privacy analysis demonstrates that PEGA successfully preserves the confidentiality of COP contents. Experimental evaluation results on several TSP datasets and KP datasets reveal that PEGA performs equivalently to the conventional GA in approximating the optimal solution.

The Knapsack Problem with Conflict Pair Constraints on Bipartite Graphs and Extensions

In this paper, we study the knapsack problem with conflict pair constraints. After a thorough literature survey on the topic, our study focuses on the special case of bipartite conflict graphs. For complete bipartite (multipartite) conflict graphs, the problem is shown to be NP-hard but solvable in pseudo-polynomial time, and it admits an FPTAS. Extensions of these results to more general classes of graphs are also presented. Further, a class of integer programming models for the general knapsack problem with conflict pair constraints is presented, which generalizes and unifies the existing formulations. The strength of the LP relaxations of these formulations is analyzed, and we discuss different ways to tighten them. Experimental comparisons of these models are also presented to assess their relative strengths. This analysis disclosed various strong and weak points of different formulations of the problem and their relationships to different types of problem data. This information can be used in designing special purpose algorithms for KPCC involving a learning component.

Combinatorial optimization methods for yarn dyeing planning

AbstractManaging yarn dyeing processes is one of the most challenging problems in the textile industry due to its computational complexity. This process combines characteristics of multidimensional knapsack, bin packing, and unrelated parallel machine scheduling problems. Multiple customer orders need to be combined as batches and assigned to different shifts of a limited number of machines. However, several practical factors such as physical attributes of customer orders, dyeing machine eligibility conditions like flotte, color type, chemical recipe, and volume capacity of dye make this problem significantly unique. Furthermore, alongside its economic aspects, minimizing the waste of natural resources during the machine changeover and energy are sustainability concerns of the problem. The contradictory nature of these two makes the planning problem multi-objective, which adds another complexity for planners. Hence, in this paper, we first propose a novel mathematical model for this scientifically highly challenging yet very practical problem from the textile industry. Then we propose Adaptive Large Neighbourhood Search (ALNS) algorithms to solve industrial-size instances of the problem. Our computational results show that the proposed algorithm provides near-optimal solutions in very short computational times. This paper provides significant contributions to flexible manufacturing research, including a mixed-integer programming model for a novel industrial problem, providing an effective and efficient adaptive large neighborhood search algorithm for delivering high-quality solutions quickly, and addressing the inefficiencies of manual scheduling in textile companies; reducing a time-consuming planning task from hours to minutes.