Multiple Knapsack Research Articles

Computation power maximization for mobile edge computing enabled dense network

High-density connection is among the natures of next-generation wireless communication systems. Meanwhile, various computation-intensive smart applications are becoming growing popularity with the technology boom. Thus, strong computation power will become a crucial requirement for wireless communication systems. Mobile edge computing provides a promising solution to this requirement by pushing a cloud-like computation capacity to the network edge. This study aims to maximize the computation power of a mobile edge computing enabled dense network. To this end, a computation bits maximization problem is formulated by jointly optimizing offloading decision and resource allocation. The problem is a mixed-variable nonlinear programming problem. First, by analyzing the feasibility of computation modes and constructing a user distribution strategy, the original problem is decoupled into two sub-problems, i.e., the assignment of the offloading users and resource allocation. The offloading users’ assignment is modeled as a restricted multiple knapsack problem, and a profit density is defined to maximize the overall profits of multiple knapsacks. Then, an improved differential evolution algorithm is developed to address the knapsack problem, in which mutation and repair operators are designed according to the profit density. Based on the optimal solution to the knapsack problem, the resource allocation sub-problem is solved by the corresponding calculation. Finally, extensive experiments are conducted to evaluate the performance of our scheme. Results show that: (1) our scheme provides superior computation power compared to that of benchmark schemes; (2) the performance gain of our scheme over benchmark schemes expands with growing connection density. Therefore, our scheme is an effective computation power optimization scheme.

QMKPy: A Python Testbed for the Quadratic Multiple Knapsack Problem

QMKPy provides a Python framework for modeling and solving the quadratic multiple knapsack problem (QMKP). It is primarily aimed at researchers who develop new solution algorithms for the QMKP. QMKPy therefore mostly functions as a testbed to quickly implement novel algorithms and compare their results with existing ones. However, the package also already includes implementations of established algorithms for those who only need to solve a QMKP as part of their research.

LP relaxation and dynamic programming enhancing VNS for the multiple knapsack problem with setup

AbstractWe consider the multiple knapsack problem (KP) with setup (MKPS), which is an extension of the KP with setup (KPS). We propose a new solving approach denoted by LP&DP‐VNS that combines linear programming (LP) relaxation and dynamic programming (DP) to enhance variable neighborhood search (VNS). The LP&DP‐VNS is tailored to the characteristics of the MKPS and reduced to LP&DP to solve the KPS. The approach is tested on 200 KPS and 360 MKPS benchmark instances. Computational experiments show the effectiveness of the LP&DP‐VNS, compared to integer programming (using CPLEX) and the best state‐of‐the‐art algorithms. It reaches 299/342 optimal solutions and 316/318 best‐known solutions and provides 127 new best solutions. An additional computational study shows that the LP&DP‐VNS scales up extremely well, solving optimally and near‐optimally very large instances with up 200 families and 300,000 items in a reasonable amount of time.

Efficient 0/1-Multiple-Knapsack Problem Solving by Hybrid DP Transformation and Robust Unbiased Filtering

The multiple knapsack problem (0/1-mKP) is a valuable NP-hard problem involved in many science-and-engineering applications. In current research, there exist two main approaches: 1. the exact algorithms for the optimal solutions (i.e., branch-and-bound, dynamic programming (DP), etc.) and 2. the approximate algorithms in polynomial time (i.e., Genetic algorithm, swarm optimization, etc.). In the past, the exact-DP could find the optimal solutions of the 0/1-KP (one knapsack, n objects) in O(nC). For large n and massive C, the unbiased filtering was incorporated with the exact-DP to solve the 0/1-KP in O(n + C′) with 95% optimal solutions. For the complex 0/1-mKP (m knapsacks) in this study, we propose a novel research track with hybrid integration of DP-transformation (DPT), exact-fit (best) knapsack order (m!-to-m2 reduction), and robust unbiased filtering. First, the efficient DPT algorithm is proposed to find the optimal solutions for each knapsack in O([n2,nC]). Next, all knapsacks are fulfilled by the exact-fit (best) knapsack order in O(m2[n2,nC]) over O(m![n2,nC]) while retaining at least 99% optimal solutions as m! orders. Finally, robust unbiased filtering is incorporated to solve the 0/1-mKP in O(m2n). In experiments, our efficient 0/1-mKP reduction confirmed 99% optimal solutions on random and benchmark datasets (n δ 10,000, m δ 100).

A Branch-and-Price Algorithm for the Multiple Knapsack Problem

The multiple knapsack problem is a well-studied combinatorial optimization problem with several practical and theoretical applications. It consists of packing some subset of n items into m knapsacks such that the total profit of the chosen items is maximum. A new formulation of the problem is presented, where a Lagrangian relaxation is derived, and we prove that it dominates the commonly used relaxations for this problem. We also present a Dantzig-Wolfe decomposition of the new formulation that we solve to optimality using a branch-and-price algorithm, where its main advantage comes from the fact that it is possible to control whether an item is included in some knapsack or not. An improved algorithm for solving the resulting packing subproblems is also introduced. Computational experiments then show that the new approach achieves state-of-the-art results. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Canadian Natural Sciences and Engineering Research Council (NSERC) [Grants 2017-06054 and 2021-04037]. This support is gratefully acknowledged.

Planning delivery-by-drone micro-fulfilment centres

Delivery drones are a disruptive technology that is spurring logistics system change, such as the adoption of urban micro-fulfilment centres (MFCs). In this paper, we develop and implement a two-stage continuum approximation (CA) model of this disruptive system in a geographic information system. The model includes common CA techniques at a local level to minimise cost, and then these local solutions are used in a second stage regional location-allocation multiple knapsack problem. We then compare the drone MFC system to a traditional delivery-by-van system and investigate potential cost or emissions savings by adjusting time-window demand, logistical sprawl, electric van alternatives, and MFC emissions. Furthermore, we conduct a sensitivity analysis to show that uncertainty in demand and effective storage density both significantly influence the number of MFCs selected and benchmark our model against commercial solvers. This methodology may also be further developed and applied to other new delivery vehicle modes.

Effect of learning strategies in an evolutionary method: the case of the bi-objective quadratic multiple knapsack problem

Effect of learning strategies in an evolutionary method: the case of the bi-objective quadratic multiple knapsack problem

A Local Search Framework for Experimental Design

We present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov's exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design.

Joint Computation Offloading and Resource Allocation Under Task-Overflowed Situations in Mobile-Edge Computing

With the rapid development of Artificial Intelligence (AI) and Internet of Things (IoT), we have to perform increasingly more resource-hungry and compute-intensive applications on IoT devices, where the available computing resources are insufficient. With the assistance of Mobile Edge Computing (MEC), offloading partial complex tasks from mobile devices to edge servers can achieve faster response time and lower energy consumption. However, it still suffers from finding the optimal offloading decision when the total amount of computations overflows the available computing resources in MEC systems. In this paper, we establish a multi-user and multi-task MEC model and design an offloading indicator, through which we analyze what the current environment belongs to. In the cases where the computational resources of devices are sufficient or partially sufficient, we utilize the relationship between the offloading indicator and the cost incurred by the tasks that are executed in the current workflow to find the optimal offloading decision. In the cases where the computation on local and edge are both insufficient, we propose a novel Offloading Algorithm based on K-means clustering and Genetic algorithm for solving Multiple knapsack problem (OAKGM), aiming not only to jointly optimize the time and energy incurred by the tasks that are executed in the current workflow, but also to penalize the overflowed computations so that the task pressure in the next workflow can be greatly reduced. In addition, a simplified Offloading Algorithm based on Multiple Knapsack Problem (OAMKP) is proposed to further cope with the environments with a large number of users or tasks. Experimental results demonstrate the effectiveness and superiority of the proposed algorithms when compared with several benchmark offloading algorithms, which can better exploit the computing capacities of IoT devices and the edge server, greatly avoid resource occupation in edge nodes and make sustainable MEC possible.

Joint Radio Resource Allocation and Beamforming Optimization for Industrial Internet of Things in Software-Defined Networking-Based Virtual Fog-Radio Access Network 5G-and-Beyond Wireless Environments

Fog computing-based radio access network (Fog-RAN) leveraging the software-defined networking (SDN) and network function virtualization (NFV) is the most promising solution to offer real-time support for the massive number of connected devices in the industrial Internet of Things (IIoT) networks. However, designing an optimal dynamic radio resource allocation to handle the fluctuating traffic loads is critical. In this article, a novel architectural design of an SDN-based virtual Fog-RAN is proposed, in which we jointly study radio resource allocation and transmit beamforming to improve resource utilization and IIoT users’ satisfaction, by minimizing the network power consumption (NPC) and maximizing the achievable sum-rate (ASR), simultaneously. To this end, we first formulate a mixed-integer nonlinear problem to optimize the physical resource block allocation, the assignment of user equipments, and radio unit, and the downlink transmit beamforming, by considering imperfect channel state information. To solve the ntractable MINLP, we exploit the successive convex approximation approach. Then, we formulate a multiple knapsack problem (MKP) to optimize the assignment between RUs and virtual baseband units, by exploiting the set of active RUs minimized in the previous problem. We solve the formulated MKP by decomposing the dual problems and solving them through the dual descent method. Through performance analysis, we show the proposed approach provides a high users’ satisfaction rate, maximizes the ASR and minimizes the NPC, and provides better savings, in terms of the number of radio and baseband resources utilized, than its counterparts.

Multi-cover inequalities for totally-ordered multiple knapsack sets: theory and computation

We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus, we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have a number of interesting properties. First, they generalize the well-known (1, k)-configuration inequalities. Second, they are not aggregation cuts. Third, they cannot be generated as rank-1 Chvátal-Gomory cuts from the inequality system consisting of the knapsack constraints and all their minimal cover inequalities. We also provide conditions under which the inequalities are facets for the convex hull of the totally-ordered knapsack set, as well as conditions for those inequalities to fully characterize its convex hull. We give an integer program to solve the separation and provide numerical experiments that showcase the strength of these new inequalities

A polynomial-time approximation scheme for parallel two-stage flowshops under makespan constraint

As a hybrid of the Parallel Two-stage Flowshop problem and the Multiple Knapsack problem, we investigate the scheduling of parallel two-stage flowshops under makespan constraint, which was motivated by applications in cloud computing and introduced by Chen et al. [3] recently. A set of two-stage jobs are selected and scheduled on parallel two-stage flowshops to achieve the maximum total profit while maintaining the given makespan constraint. We give a positive answer to an open question about its approximability proposed by Chen et al. [3]. More specifically, based on guessing strategies and rounding techniques for linear programs, we present a polynomial time approximation scheme (PTAS) for the case when the number of flowshops is a fixed constant. As the case with two flowshops is already strongly NP-hard, our PTAS achieves the best possible approximation ratio.

Deep Optimisation: Transitioning the Scale of Evolutionary Search by Inducing and Searching in Deep Representations

We investigate the optimisation capabilities of an algorithm inspired by the Evolutionary Transitions in Individuality. In these transitions, the natural evolutionary process is repeatedly rescaled through successive levels of biological organisation. Each transition creates new higher-level evolutionary units that combine multiple units from the level below. We call the algorithm Deep Optimisation (DO) to recognise both its use of deep learning methods and the multi-level rescaling of biological evolutionary processes. The evolutionary model used in DO is a simple hill-climber, but, as higher-level representations are learned, the hill-climbing process is repeatedly rescaled to operate in successively higher-level representations. The transition process is based on a deep learning neural network (NN), specifically a deep auto-encoder. Our experiments with DO start with a study using the NP-hard problem, multiple knapsack (MKP). Comparing with state-of-the-art model-building optimisation algorithms (MBOAs), we show that DO finds better solutions to MKP instances and does so without using a problem-specific repair operator. A second, much more in-depth investigation uses a class of configurable problems to understand more precisely the distinct problem characteristics that DO can solve that other MBOAs cannot. Specifically, we observe a polynomial vs exponential scaling distinction where DO is the only algorithm to show polynomial scaling for all problems. We also demonstrate that some problem characteristics need a deep network in DO. In sum, our findings suggest that the use of deep learning principles have significant untapped potential in combinatorial optimisation. Moreover, we argue that natural evolution could be implementing something like DO, and the evolutionary transitions in individuality are the observable result.

Scheduling multiple two-stage flowshops with a deadline

Recently, motivated by applications in cloud computing, scheduling multiple two-stage flowshops with the objective of minimizing the makespan has drawn increasing attention. Motivated by the same applications, the current paper studies the problem of scheduling multiple two-stage flowshops with a deadline, with the objective of maximizing the profit of the jobs that can be completed by the deadline. Algorithms and complexity of the problem are investigated. For the case where the number of two-stage flowshops is part of the input, we present an efficient approximation algorithm with a constant ratio. The approximation ratio is improved via a study of the relationship between the problem and the multiple knapsack problem, combined with the recent progresses in the research in approximation algorithms for the multiple knapsack problem. By integrating techniques in the study of the classical Knapsack problem and the classical Makespan problem on multiple processors, plus additional new techniques, a polynomial-time approximation algorithm with a further improved approximation ratio is developed for the case where the number of flowshops is a fixed constant.

A Deep Reinforcement Learning-Based Scheme for Solving Multiple Knapsack Problems

A knapsack problem is to select a set of items that maximizes the total profit of selected items while keeping the total weight of the selected items no less than the capacity of the knapsack. As a generalized form with multiple knapsacks, the multi-knapsack problem (MKP) is to select a disjointed set of items for each knapsack. To solve MKP, we propose a deep reinforcement learning (DRL) based approach, which takes as input the available capacities of knapsacks, total profits and weights of selected items, and normalized profits and weights of unselected items and determines the next item to be mapped to the knapsack with the largest available capacity. To expedite the learning process, we adopt the Asynchronous Advantage Actor-Critic (A3C) for the policy model. The experimental results indicate that the proposed method outperforms the random and greedy methods and achieves comparable performance to an optimal policy in terms of the profit ratio of the selected items to the total profit sum, particularly when the profits and weights of items have a non-linear relationship such as quadratic forms.

Effect of the local branching strategy on the descent method: The case of the generalized multiple knapsack with setup

In this paper, we investigate the use of the local branching strategy into an iterative descent method for tackling the generalized multiple knapsack problem with setup. An instance of the problem is composed of a set of classes containing items, and a set of available knapsacks. Its objective consists in selecting the subsets of items belonging to the classes with a maximum objective value: each item is characterized with its profit and weight while a class is characterized with its cost such that a given item may be selected whenever its corresponding class is activated, and an item can be configured (setup) for a single knapsack. The proposed iterative method starts by solving a series of reduced subproblems built by adding a series of cardinality constraints. Next, the local branching strategy is iteratively employed for highlighting the efficiency of the method. Finally, the behavior of the method is evaluated on a set of instances extracted from the literature, where its achieved bounds are compared to those reached by the best methods available in the literature. A statistical analysis is also provided showing the high performance of the method. New bounds are discovered.

An almost optimal approximation algorithm for monotone submodular multiple knapsack

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set I of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a packing of a subset of items A⊆I in the bins such that f(A) is maximized. Our main result is an almost optimal polynomial time (1−e−1−ε)-approximation algorithm for the problem, for any ε>0. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints.

Multiplicative knapsack injectivity as condition of effective unauthorized access protection

The mathematical modelling of data protection systems based on multiplicative knapsack is considered. In particular, development of cryptographic solution to hierarchic access control problem is represented. Previously in research the authors has formulated the requirements to knapsack in order to apply these to a hierarchic access control problem based on the requirements for similar models. Here in the article the possibility of practical implementation of the construction of such knapsacks is pointed. Based on special mathematical model for multiplicative knapsacks generalized sufficient condition of multiplicative knapsack injectivity is formulated and proved in the article. Known sufficient conditions of multiplicative knapsack injectivity are special cases of authors condition. The mathematical apparatus of additive knapsacks for building multiplicative vector is shown.

A hybrid population-based algorithm for the bi-objective quadratic multiple knapsack problem

In this paper, the bi-objective quadratic multiple knapsack problem is tackled with a hybrid population-based method. The proposed method starts by computing two reference solutions, where a specialized powerful mono-objective algorithm is used. From both reference solutions, a starting population is built by using a series of perturbations around the solutions. Next, the so-called non-sorting genetic process is combined with a new drop/rebuild operator for generating a series of populations till converging toward an approximate Pareto front with high density. The performance of the hybrid population based algorithm (namely HBPA) is evaluated on a set of benchmark instances of the literature containing both medium and large-scale instances. Its provided results are compared to those achieved by the best methods available in the literature. Encouraging results have been obtained.

Competitive Algorithms for the Online Multiple Knapsack Problem with Application to Electric Vehicle Charging

We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. We introduce a new algorithm that achieves a competitive ratio within an additive factor of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. Finally, in the full version of this paper, we illustrate the proposed algorithm via trace-based experiments of EV charging.