Knapsack Problem Research Articles

GCKSign: Simple and efficient signatures from generalized compact knapsack problems.

In 2009, Lyubashevsky proposed a lattice-based signature scheme using the Schnorr-like identification and the Fiat-Shamir heuristic and proved its security under the collision resistance of a generalized compact knapsack function. However, their security analysis requires the witness indistinguishability property, leading to significant inefficiency and an increase of sizes of public key and signature. To overcome the efficiency issue associated with the WI property, we introduce a new lattice-based assumption, called the target-modified one-wayness problem of the GCK function and show its reduction to well-known lattice-based problems. Additionally, we present a simple and efficient GCK-based signature scheme, GCKSign, whose security is based on the Module GCK-TMO problem in the random oracle model. GCKSign is a natural extension of Lyubashevsky's scheme in a module setting, but achieves considerable efficiency gains due to eliminating the witness indistinguishability property. As a result, GCKSign achieves approximately 3.4 times shorter signature size and 2.4 times shorter public key size at the same security level.

A Framework for 5G Network Slicing Optimization using 2-Edge-Connected Subgraphs for Path Protection

Emerging telecommunications technologies require robust frameworks for efficient network slicing. We propose a network-slicing model that aims to optimize the deployment of virtual networks on a physical network topology. Our model ensures compliance with 5G requirements, incorporating latency and capacity constraints on virtual links. Selecting slices with cost and resource requirements on the computing nodes is optimized using a Knapsack problem with revenue maximization. We propose a path protection algorithm to deal with link failures by constructing a 2-edge-connected subgraph (or two link-disjoint Steiner trees) for each slice to provide both primary and backup paths. Simulation results include comparison with existing solutions by metrics such as latency, revenue, resource utilization, number of protected slices, and computation time, providing valuable insights for network planners operating in diverse and dynamic environments. Key contributions include efficient resource allocation using the Knapsack problem, enhanced network resilience via 2-edge-connected subgraphs for path protection, and realistic simulation experiments on SNDlib dataset topologies. The simulation results show that the proposed framework improves computational efficiency compared to the recent related solutions, particularly in large network topologies where k-connected function slicing (KC- FS) subgraph embeddings take approximately 3.5 times more computation time.

An effective binary dynamic grey wolf optimization algorithm for the 0-1 knapsack problem

An effective binary dynamic grey wolf optimization algorithm for the 0-1 knapsack problem

Correction to: An integrated ILS-VND strategy for solving the knapsack problem with forfeits

Correction to: An integrated ILS-VND strategy for solving the knapsack problem with forfeits

Charging Scheduling Method for Wireless Rechargeable Sensor Networks Based on Energy Consumption Rate Prediction for Nodes.

With the development of the IoT, Wireless Rechargeable Sensor Networks (WRSNs) derive more and more application scenarios with diverse performance requirements. In scenarios where the energy consumption rate of sensor nodes changes dynamically, most existing charging scheduling methods are not applicable. The incorrect estimation of node energy requirement may lead to the death of critical nodes, resulting in missing events. To address this issue, we consider both the spatial imbalance and temporal dynamics of the energy consumption of the nodes, and minimize the Event Missing Rate (EMR) as the goal. Firstly, an Energy Consumption Balanced Tree (ECBT) construction method is proposed to prolong the lifetime of each node. Then, we transform the goal into Maximizing the value of the Evaluation function of each node's Energy Consumption Rate prediction (MEECR). Afterwards, the setting of the evaluation function is explored and the MEECR is further transformed into a variant of the knapsack problem, namely "the alternating backpack problem", and solved by dynamic programming. After predicting the energy consumption rate of the nodes, a charging scheduling scheme that meets the Dual Constraints of Nodes' energy requirements and MC's capability (DCNM) is developed. Simulations demonstrate the advantages of the proposed method. Compared to the baselines, the EMR was reduced by an average of 35.2% and 26.9%.

Two-set inequalities for the binary knapsack polyhedra

This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.

Mathematical models based on decision hypergraphs for designing a storage cabinet

We study the problem of designing a cabinet made up of a set of shelves that contain compartments whose contents slide forward on opening. Considering a set of items candidate to be stored in the cabinet over a given time horizon, the problem is to design a set of shelves and a set of compartments on each shelf, and select the items to insert into the compartments. The objective is to maximize the sum of the profits of the selected items. We call our problem the Storage Cabinet Physical Design (SCPD) problem. The SCPD problem combines a two-staged two-dimensional knapsack problem for designing the shelves and compartments with a set of temporal knapsack problems for selecting and assigning items to compartments. We formalize the SCPD problem and formulate it as a maximum cost flow problem in a decision hypergraph with additional linear constraints. To reduce the size of this model, we break symmetries, generalize graph compression techniques and exploit dominance rules for precomputing subproblem solutions. We also present a set of valid inequalities to improve the linear relaxation of the model. We empirically show that solving the arc-flow model with our enhancements outperforms solving a compact mixed integer linear programming formulation of the SCPD problem.

A quantum artificial bee colony algorithm based on quantum walk for the 0–1 knapsack problem

Based on the characteristics of the quantum mechanism, a novel quantum walk artificial bee colony algorithm is proposed to promote performance. Firstly, the discrete quantum walk is an approach taken to search for new food sources in the updated phase for employed bees and onlooker bees, which can enhance the probability of the target solution to extend the exploration capability. Secondly, the food source selection policy of the onlooker bees changes, from roulette selection to tournament selection, to boost exploitation and convergence speed. Finally, the novel algorithm is brought forward, along with the approach to analyze 0–1 knapsack problems. The experimental results prove that our algorithm can overcome the premature phenomenon and perform better in the areas of search capability, convergence speed, and stability performance. The performance is superior to that of the conventional artificial bee colony algorithm, as well as the genetic algorithm, in a set of 0–1 knapsack problems.

An optimization framework for solving large scale multidemand multidimensional knapsack problem instances employing a novel core identification heuristic

By applying the core concept to solve a binary integer program (BIP), certain variables of the BIP are fixed to their anticipated values in the optimal solution. In contrast, the remaining variables, called core variables, are used to construct and solve a core problem (CP) instead of the BIP. A new approach for identifying CP utilizing a local branching (LB) alike constraint is presented in this article. By including the LB-like constraint in the linear programming relaxation of the BIP, this method transfers batches of variables to the set of core variables by analyzing changes to their reduced costs. This approach is sensitive to problem hardness because more variables are moved to the core set for hard problems compared to easy ones. This novel core identification approach is embedded in a multi-stage framework to solve the multidemand, multidimensional knapsack problems (MDMKP), where at each stage, more variables are added to the previous stage CP. The default branch and bound of CPLEX20.10 is used to solve the first stage, and a tabu search algorithm is used to solve subsequent stages until all variables are added to CP in the last stage. The new framework has shown equivalent to superior results compared to the state-of-the-art algorithms in solving large MDMKP instances having 500 and 1,000 variables.

A self-adaptive arithmetic optimization algorithm with hybrid search modes for 0–1 knapsack problem

A self-adaptive arithmetic optimization algorithm with hybrid search modes for 0–1 knapsack problem

A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints

A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints

DELTA: Memory-Efficient Training via Dynamic Fine-Grained Recomputation and Swapping

To accommodate the increasingly large-scale models within limited-capacity GPU memory, various coarse-grained techniques, such as recomputation and swapping, have been proposed to optimize memory usage. However, these methods have encountered limitations, either in terms of inefficient memory reduction or diminished training performance. In response to this, our paper introduces DELTA, an innovative approach for memory-efficient large-scale model training that combines fine-grained memory optimization and prefetching technology to reduce memory usage while maintaining high training throughput concurrently. Initially, we formulate the problem of memory-throughput joint optimization as an easy-solving 0/1 Knapsack problem. Leveraging this formalization, we use an improving polynomial complexity heuristic algorithm to address the problem effectively. Furthermore, we introduce a novel bidirectional prefetching technology into dynamic memory management, which significantly accelerates the model training when compared to relying solely on recomputation or swapping. Finally, DELTA offers users an automated training execution library, eliminating the need for manual configuration or specialized expertise. Experimental results demonstrate the effectiveness of DELTA in reducing GPU memory consumption. Compared to state-of-the-art methods, DELTA achieves substantial memory savings ranging from 40% to 72%, while maintaining comparable convergence performance for various models, including ResNet-50, ResNet-101, and BERT-Large. Notably, DELTA enables the training of GPT2-Large and GPT2-XL with batch sizes increased by 5.5 × and 6 ×, respectively, showcasing its versatility and practicality in enabling large-scale model training on GPU hardware.

Load-aware switch migration for controller load balancing in edge–cloud architectures

As the fundamental infrastructure for edge–cloud architectures, the inter-datacenter elastic optical network is used for data analysis and processing. As the demand for applications increases, the large number of service requests increases the processing overhead in the control plane, resulting in unbalanced controller loads. Existing switch migration mechanisms have been proposed for controller load balancing. Unfortunately, most of the existing mechanisms only consider the switch with the highest flow request rate as the migration object in the process of switch selection, and ignore the migration cost generated in the switch migration activity, such as the update cost of flow request message and the deployment cost of migration rule, which may increase the controller load. Additionally, most of them choose the controller with light load as the target controller to associate with the switch to be migrated, without considering whether the target controller is overloaded after the switch migration, which leads to the low load balancing performance of the controller. In view of the above problems, this paper proposes a Load-Aware Switch Migration (LASM) mechanism in edge–cloud architectures. The LASM mechanism models and analyses the cost metrics affecting switch migration and selects lower-cost switches from overloaded controller-controlled domain networks for migration activities. Besides, the LASM mechanism models switch migration based on the 0-1 knapsack problem and avoids overloading the target controllers through a greedy policy to achieving optimal migration activities. The experimental results show that the proposed LASM mechanism improves controller load balancing performance by an average of 34.3%, eliminates migration costs by 30.2%, and reduces response times by an average of 39.3%, respectively, compared to existing solutions.

Enhancing computational efficiency in solving Knapsack problem: insights from algorithic parallelization and optimization

Enhancing computational efficiency in solving Knapsack problem: insights from algorithic parallelization and optimization

Continuous Equality Knapsack with Probit-Style Objectives

We study continuous, equality knapsack problems with uniform separable, non-convex objective functions that are continuous, antisymmetric about a point, and have concave and convex regions. For example, this model captures a simple allocation problem with the goal of optimizing an expected value where the objective is a sum of cumulative distribution functions of identically distributed normal distributions (i.e., a sum of inverse probit functions). We prove structural results of this model under general assumptions and provide two algorithms for efficient optimization: (1) running in linear time and (2) running in a constant number of operations given preprocessing of the objective function.

An integrated ILS-VND strategy for solving the knapsack problem with forfeits

An integrated ILS-VND strategy for solving the knapsack problem with forfeits

A bay design problem in less-than-unit-load production warehouse

In this paper, we consider a bay design problem in a less-than-unit-load production warehouse, which is motivated by a real-world problem in a semiconductor company. The objective is to maximize the utilization of the vertical space in bays by considering several practical storage requirements. To solve the problem, a non-linear integer programming model is first formulated. Since the problem is similar to a two-stage cutting stock problem (CSP), a column-and-row generation (CRG) method is developed, in which the original problem is decomposed into a restricted master problem and three subproblems, including two classical column generation subproblems and a row generation subproblem. The two former subproblems are solved as unbounded knapsack problems and for the latter, a two-stage approach is applied. The results of computational experiments on randomly generated instances show that the proposed CRG method is more efficient than the classic column-generation-based method, solving the non-linear model directly and solving a cut model in the literature directly. The results of a case study show that our strategy can improve the utilization of the existing warehouse storage space significantly by about 24%. The CRG method is also tested on basic two-stage two-dimensional CSP benchmarks and its performance is compared to those of other pattern-based methods. The results show its potential for effectively solving the basic two-stage two-dimensional CSPs.

A biased random-key genetic algorithm for the knapsack problem with forfeit sets

This work addresses the Knapsack Problem with Forfeit Sets, a recently introduced variant of the 0/1 Knapsack Problem considering subsets of items associated with contrasting choices. Some penalty costs need to be paid whenever the number of items in the solution belonging to a forfeit set exceeds a predefined allowance threshold. We propose an effective metaheuristic to solve the problem, based on the Biased Random-Key Genetic Algorithm paradigm. An appropriately designed decoder function assigns a feasible solution to each chromosome, and improves it using some additional heuristic procedures. We show experimentally that the algorithm outperforms significantly a previously introduced metaheuristic for the problem.

Towards Collaborative Edge Intelligence: Blockchain-Based Data Valuation and Scheduling for Improved Quality of Service

Collaborative edge intelligence, a distributed computing paradigm, refers to a system where multiple edge devices work together to process data and perform distributed machine learning (DML) tasks locally. Decentralized Internet of Things (IoT) devices share knowledge and resources to improve the quality of service (QoS) of the system with reduced reliance on centralized cloud infrastructure. However, the paradigm is vulnerable to free-riding attacks, where some devices benefit from the collective intelligence without contributing their fair share, potentially disincentivizing collaboration and undermining the system’s effectiveness. Moreover, data collected from heterogeneous IoT devices may contain biased information that decreases the prediction accuracy of DML models. To address these challenges, we propose a novel incentive mechanism that relies on time-dependent blockchain records and multi-access edge computing (MEC). We formulate the QoS problem as an unbounded multiple knapsack problem at the network edge. Furthermore, a decentralized valuation protocol is introduced atop blockchain to incentivize contributors and disincentivize free-riders. To improve model prediction accuracy within latency requirements, a data scheduling algorithm is given based on a curriculum learning framework. Based on our computer simulations using heterogeneous datasets, we identify two critical factors for enhancing the QoS in collaborative edge intelligence systems: (1) mitigating the impact of information loss and free-riders via decentralized data valuation and (2) optimizing the marginal utility of individual data samples by adaptive data scheduling.

0-1 Knapsack Problem Solving using Genetic Optimization Algorithm

0-1 Knapsack Problem Solving using Genetic Optimization Algorithm