Knapsack Problem Research Articles

List-Based Threshold Accepting Algorithm with Improved Neighbor Operator for 0–1 Knapsack Problem

The list-based threshold accepting (LBTA) algorithm is a sophisticated local search method that utilizes a threshold list to streamline the parameter tuning process in the traditional threshold accepting (TA) algorithm. This paper proposes an enhanced local search version of the LBTA algorithm specifically tailored for solving the 0–1 knapsack problem (0–1 KP). To maintain a dynamic threshold list, a feasible threshold updating strategy is designed to accept adaptive modifications during the search process. In addition, the algorithm incorporates an improved bit-flip operator designed to generate a neighboring solution with a controlled level of disturbance, thereby fostering exploration within the solution space. Each trial solution produced by this operator undergoes a repair phase using a hybrid greedy repair operator that incorporates both density-based and value-based add operator to facilitate optimization. The LBTA algorithm’s performance was evaluated against several state-of-the-art metaheuristic approaches on a series of large-scale instances. The simulation results demonstrate that the LBTA algorithm outperforms or is competitive with other leading metaheuristics in the field.

The quadratic knapsack problem with setup

The Quadratic Knapsack Problem is a well-known generalization of the classical 0-1 knapsack problem, in which any pair of items produces a pairwise profit if both are selected. Another relevant generalization of the knapsack problem is the Knapsack Problem with Setup, in which the items are partitioned into classes, the items of a class can only be inserted into the knapsack if the corresponding class is activated, and activating a class involves a setup cost and a setup capacity reduction.Despite a rich literature on these two problems, their obvious generalization, i.e., the Quadratic Knapsack Problem with Setup, was never investigated so far. We discuss applications, mathematical models, deterministic matheuristic algorithms, and computationally evaluate their performance.

Branch-and-Bound and Dynamic Programming Approaches for the Knapsack Problem

Branch-and-Bound and Dynamic Programming Approaches for the Knapsack Problem

Expectation analysis for bounding solutions of the 0–1 knapsack problem

In this paper, an entirely novel discrete probabilistic model is presented to generate 0–1 Knapsack Problem instances. We analyze the expected behavior of the greedy algorithm, the eligible-first algorithm and the linear relaxation algorithm for these instances; all used to bound the solution of the 0–1 Knapsack Problem (0–1 KP) and/or its approximation. The probabilistic setting is given and the main random variables are identified. The expected performance for each of the aforementioned algorithms is analytically established in closed forms in an unprecedented way.

Approaches for the On-Line Three-Dimensional Knapsack Problem with Buffering and Repacking

The rapid growth of the e-commerce sector, particularly in Latin America, has highlighted the need for more efficient automated packing and distribution systems. This study presents heuristic algorithms to solve the online three-dimensional knapsack problem (OSKP), incorporating buffering and repacking strategies to optimize space utilization in automated packing environments. These strategies enable the system to handle the stochastic nature of item arrivals and improve container utilization by temporarily storing boxes (buffering) and rearranging already packed boxes (repacking) to enhance packing efficiency. Computational experiments conducted on specialized datasets from the existing literature demonstrate that the proposed heuristics perform comparably to state-of-the-art methodologies. Moreover, physical experiments were conducted on a robotic packing cell to determine the time that buffering and repacking implicate. The contributions of this paper lie in the integration of buffering and repacking into the OSKP, the development of tailored heuristics, and the validation of these heuristics in both simulated and real-world environments. The findings indicate that including buffering and repacking strategies significantly improves space utilization in automated packing systems. However, they significantly increase the time spent packing.

Quantum Algorithm for Maximum Integer M-Dimensional Knapsack Problem by Shor’s Fourier Transform with RAM on QCEngine

Quantum Algorithm for Maximum Integer M-Dimensional Knapsack Problem by Shor’s Fourier Transform with RAM on QCEngine

Knapsack Balancing via Multiobjectivization

In this paper, we address the aspect of knapsack balancing in the classic knapsack problem. Recognizing that excessive dispersion in the objective function or constraint coefficients of the optimal solution can be undesirable, we propose, when appropriate, to control this effect through problem multiobjectivization. By multiobjectivization, we mean the addition of one or more objective functions that aim to shift the original problem’s optimal solutions towards Pareto optimal solutions of the multiobjectivized problem, reducing the dispersion of the respective coefficients. We detail how the knapsack balance aspect can be incorporated into the standard knapsack problem model and demonstrate the functionality of this enriched model through illustrative examples.

Using Online Algorithms to Solve NP-Hard Problems

This paper explores the application of online algorithms to tackle NP-hard problems, a class of computational challenges characterized by their intractability and wide-ranging real-world implications. Unlike traditional offline algorithms that have access to complete input data, online algorithms make decisions sequentially, often under constraints of incomplete information. We investigate various strategies, including greedy approaches, randomization, and competitive analysis, to assess their effectiveness in solving NP-hard problems such as the Traveling Salesman Problem, the Knapsack Problem, and the Set Cover Problem. Our analysis highlights the trade-offs between solution quality and computational efficiency, emphasizing the significance of the competitive ratio in evaluating algorithm performance. Additionally, we discuss the practical applications of online algorithms in dynamic environments, such as real-time systems and streaming data processing. Through a comprehensive review of existing literature and novel algorithmic designs, we aim to provide insights into the viability of online algorithms as a robust framework for addressing NP-hard problems in scenarios where immediate decision-making is crucial. The findings underscore the potential for future research to enhance these algorithms, making them increasingly applicable in complex, real-world contexts.

THE BRANCH AND BOUND APPROACH TO A BOUNDED KNAPSACK PROBLEM (CASE STUDY: OPTIMIZING OF PENCAK SILAT MATCH SESSIONS)

A method commonly employed to solve integer programming problems is the Branch and Bound. In this article, maximizing the number of matches held on the first day of pencak silat tournaments is essential because it can impact the overall dynamics and results of the competition. The model used to maximize the number of match sessions in pencak silat competitions is a variant of the Bounded Knapsack Problem (BKP), belonging to the category of integer programming models. The result obtained using the Branch and Bound method ensures that the maximum number of match sessions can be conducted. The objective value obtained using the Branch and Bound method decreases as it descends, indicating a decreasing maximum value.

Algorithms for the thief orienteering problem on directed acyclic graphs

We consider the scenario of routing an agent called a thief through a weighted graph G=(V,E) from a start vertex s to an end vertex t. A set I of items each with weight wi and profit pi is distributed among V∖{s,t}. The thief, who has a knapsack of capacity W, must follow a simple path from s to t within a given time T while packing in the knapsack a set of items taken from the vertices along the path of total weight at most W and maximum profit. The travel time across an edge depends on the edge length and current knapsack load.The thief orienteering problem (ThOP) is a generalization of the orienteering problem, the longest path problem, and the 0-1 knapsack problem. We prove that there exists no approximation algorithm for ThOP with constant approximation ratio unless P=NP, and we present a polynomial-time approximation scheme (PTAS) for a relaxed version of ThOP when G is directed and acyclic that produces solutions that use time at most T(1+ϵ) for any constant ϵ>0. We also present a fully polynomial-time approximation scheme (FPTAS) for ThOP on arbitrary undirected graphs where the travel time depends only on the lengths of the edges and T is the length of a shortest path from s to t plus a constant K. Finally, we present a FPTAS for a restricted version of the problem where the input graph is a clique.

CMSA based on set covering models for packing and routing problems

AbstractMany packing, routing, and knapsack problems can be expressed in terms of integer linear programming models based on set covering. These models have been exploited in a range of successful heuristics and exact techniques for tackling such problems. In this paper, we show that integer linear programming models based on set covering can be very useful for their use within an algorithm called “Construct, Merge, Solve & Adapt”(CMSA), which is a recent hybrid metaheuristic for solving combinatorial optimization problems. This is because most existing applications of CMSA are characterized by the use of an integer programming solver for solving reduced problem instances at each iteration. We present applications of CMSA to the variable-sized bin packing problem and to the electric vehicle routing problem with time windows and simultaneous pickups and deliveries. In both applications, CMSA based on a set covering model strongly outperforms CMSA when using an assignment-type model. Moreover, state-of-the-art results are obtained for both considered optimization problems.

Optimizing Knapsack Problem with ImprovedSHLO Variations

Optimizing Knapsack Problem with ImprovedSHLO Variations

Statistical analyses of solution methods for the multiple-choice knapsack problem with setups: Implications for OR practitioners

An interesting extension of the classic Knapsack Problem (KP) is the Multiple-Choice Knapsack Problem with Setups (MCKS) which is focused on solving practical applications that involve both multiple periods and setups. Sophisticated solution methods for the MCKS that are presented in the operations research (OR) literature are not readily available for use by OR practitioners. Using MCKS test instances that appear in the literature, we demonstrate that the general-purpose integer programming software Gurobi sometimes used in an iterative manner can efficiently solve these MCKS instances using all default parameter values on a standard PC. It is shown both empirically and statistically that these Gurobi solutions are competitive with solution approaches from the literature. Hence, our approach using Gurobi is both easy for the OR practitioner to use and gives results competitive with the best specialized MCKS solution methods in the literature without the need to generate algorithm-specific code. Furthermore, this paper presents significant concerns regarding the solutions stated in the literature by the approximate solution method that reports the best results on 120 MCKS test instances. Specifically, 26% of this method’s solutions violate Gurobi upper bounds and an additional 33% of its solutions, on average, exceed the known guaranteed optimums by a value of 12,510.

Solving Transport Infrastructure Investment Project Selection and Scheduling Using Genetic Algorithms

The development of transport infrastructure is crucial for economic growth, social connectivity, and sustainable development. Many countries have historically underinvested in transport infrastructure, necessitating more efficient strategic planning in the implementation of transport infrastructure investment projects. This article addresses the selection and scheduling of transport infrastructure projects, specifically within the context of utilizing pre-allocated funds within a multi-annual budget investment program. The current decision-making process relies heavily on expert judgment and lacks quantitative decision support methods. We propose a genetic algorithm as a decision-support tool, framing the problem as an NP-hard 0–1 multiple knapsack problem. The proposed genetic algorithm (GA) is unique for its matrix-encoded chromosomes, specially designed genetic operators, and a customized repair operator to address the large number of invalid chromosomes generated during the GA computation. In computational experiments, the proposed GA is compared to an exact solution and proves to be efficient in terms of quality of obtained solutions and computational time, with an average computational time of 108 s and the quality of obtained solutions typically ranging between 85% and 95% of the optimal solution. These results highlight the potential of the proposed GA to enhance strategic decision-making in transport infrastructure development.

The neural dynamics associated with computational complexity.

Many everyday tasks require people to solve computationally complex problems. However, little is known about the effects of computational hardness on the neural processes associated with solving such problems. Here, we draw on computational complexity theory to address this issue. We performed an experiment in which participants solved several instances of the 0-1 knapsack problem, a combinatorial optimization problem, while undergoing ultra-high field (7T) functional magnetic resonance imaging (fMRI). Instances varied in computational hardness. We characterize a network of brain regions whose activation was correlated with computational complexity, including the anterior insula, dorsal anterior cingulate cortex and the intra-parietal sulcus/angular gyrus. Activation and connectivity changed dynamically as a function of complexity, in line with theoretical computational requirements. Overall, our results suggest that computational complexity theory provides a suitable framework to study the effects of computational hardness on the neural processes associated with solving complex cognitive tasks.

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%.