Fractional Knapsack Research Articles

Integer optimization with penalized fractional values: The Knapsack case

We consider integer optimization problems where variables can potentially take fractional values, but this occurrence is penalized in the objective function. This general situation has relevant examples in scheduling (preemption), routing (split delivery), cutting and telecommunications, just to mention a few. However, the general case in which variables integrality can be relaxed at cost of introducing a general penalty was not discussed before. As a case study, we consider the possibly simplest combinatorial optimization problem, namely the classical Knapsack Problem. We introduce the Fractional Knapsack Problem with Penalties (FKPP), a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity. We analyze relevant properties of the problem, present alternative mathematical models, and analyze their performance from a theoretical viewpoint. In addition, we introduce a Fully Polynomial Time Approximation Scheme for the approximate solution of the general problem, and an improved dynamic programming approach that computes the optimal solution in one relevant case. We computationally test the proposed models and algorithms on a large set of instances derived from benchmarks from the literature.

Signaling in Online Retail: Efficacy of Public Signals

We consider the problem of inventory and demand signaling in online retail. Customers seek to purchase an item with a finite inventory in one of two time periods. They are strategic and Bayesian but do not know the current inventory or number of other customers. The firm knows these values and seeks to persuade customers to buy in the first time period where the price is higher. We adopt the framework of Bayesian persuasion to formulate and analyze the firm's choice of a revenue-optimal signaling mechanism. We observe that the randomness in the number of customers causes each customer to believe the demand to be stochastically higher. Taking this into consideration, we establish that when customers are homogeneous, the optimal signaling mechanism is a public mechanism where all customers are provided the same information. In particular, the firm does not benefit from providing different information to different customers, such as through personalized email marketing. We exploit the public structure of the optimal mechanism to cast the firm's problem as a fractional knapsack problem. We support our analytical results with numerical computations that show public signaling achieving substantial revenue gains over no information or full information sharing.

Multi-Targeted Downlink Scheduling for Overload-States in LTE Networks: Proportional Fractional Knapsack Algorithm With Gaussian Weights

In Long-Term Evolution (LTE) and beyond systems, radio resource scheduling mechanism plays one of the main roles in system performance maximization. From this perspective, due to the conflicting quality requirements of different traffic types, providing a compromise among all performance targets for heterogeneous traffic is difficult. Moreover, the centralized scheduling mechanism for the ever growing number of users along with the massive variety of services, especially in overload states, is infeasible due to the extensive cost of information acquisition and computations. In this paper, we design resource scheduling policies for supporting the efficient delivery of heterogeneous traffic in overload states of a cell. To this end, we cast the class-based bearer-level resource distribution problem as a Proportional Fractional Knapsack model. The objective of the formulated problem is to meet Quality of Service (QoS) requirements and provide fairness for all standardized service classes. Since the solution of this problem is computationally expensive, due to the uncertainty and limited information on network and user operation, we develop a Gaussian-based analytical model and drive a formula for simplified computation of the weight of service bearers. Then, we propose Proportional Fractional Knapsack algorithm for guaranteeing effective utilization of resources for heterogeneous traffic. Finally, performance evaluation results are provided and demonstrate that the proposed scheduling approach can provide a significant level of fairness, in balance with the QoS and throughput performance targets, comparable with optimal ones.

A Priority-Based Bandwidth Management Method in Public Safety Networks

After Sewol ferry disaster occurred in April 2014, Korean government began investing to deploy LTE-based public safety network until the year of 2017. In order to reduce the operating and capital costs, resource sharing scheme among public safety network and commercial LTE networks is considered as one of the viable approaches. This thesis proposes a method of allocating bandwidth of public safety network based on various priorities required for disaster scenarios and stages in a resource sharing environment. In order to obtain the highest efficiency, we formulate the bandwidth allocation problem as a Fractional Knapsack Problem. Greedy algorithm was applied to solve the problem. For performance evaluation, we created several disaster scenarios and set suitable parameters for each scenario based on a disaster manual. The proposed method is compared with two typical methods, which are Class-based bandwidth allocation and Uniform bandwidth allocation. The results showed that the better performance in terms of the sum of the values and the amount of lost bytes.

Greedy–knapsack algorithm for optimal downlink resource allocation in LTE networks

The Long Term Evolution (LTE) as a mobile broadband technology supports a wide domain of communication services with different requirements. Therefore, scheduling of all flows from various applications in overload states in which the requested amount of bandwidth exceeds the limited available spectrum resources is a challenging issue. Accordingly, in this paper, a greedy algorithm is presented to evaluate user candidates which are waiting for scheduling and select an optimal set of the users to maximize system performance, without exceeding available bandwidth capacity. The greedy-knapsack algorithm is defined as an optimal solution to the resource allocation problem, formulated based on the fractional knapsack problem. A compromise between throughput and QoS provisioning is obtained by proposing a class-based ranking function, which is a combination of throughput and QoS related parameters defined for each application. The simulation results show that the proposed method provides high performance in terms of throughput, loss and delay for different classes of QoS over the existing ones, especially under overload traffic.

Genetic Algorithm based Approach to Solve Non Fractional (0/1) Knapsack Optimization Problem

this paper we solve the non fractional knapsack problem also known as 0-1 knapsack using genetic algorithm. The usual approaches are greedy method and dynamic programming. Its an optimization problem where we try to maximize the values that can be put into a knapsack under the constraint of its weight. We solve the problem using genetic algorithm in matlab using gatool. In this research work different selection schemes have been used like roulette wheel, tournament selection, Stochastic selection etc. Following the introduction of genetic algorithm and knapsack problem, formulation of 0-1 knapsack problem in genetic algorithm is presented. Experimental results using various selection schemes have been analyzed and comparison of genetic algorithm technique is done with greedy method and dynamic programming optimizing techniques.

Dynamic Orthogonal Frequency Division Multiple Access resource management for downlink interference avoidance in two‐tier networks

SUMMARYIn two‐tier networks, which consist of macrocells and femtocells, femtocells can offload the traffic from macrocells thereby improving indoor signal coverage. However, the dynamic deployment feature of femtocells may result in signal interference due to limited frequency spectrum. The tradeoff between broad signal coverage and increased signal interference deserves further exploration for practical network operation. In this paper, dynamic frequency resource management is proposed to avoid both co‐tier and cross‐tier Orthogonal Frequency Division Multiple Access downlink interference and increase frequency channel utilization under co‐channel deployment. A graph‐based non‐conflict group discovery algorithm is proposed to discover the disjoint interference‐free groups among femtocells in order to avoid the co‐tier interference. A macrocell uses the femtocell gateway for frequency resource allocation among femtocells to avoid cross‐tier interference. We formulate the optimized frequency resource assignment as a fractional knapsack problem and solve the problem by using a greedy method. The simulation results show that the average data transfer rate can be increased from 21% to 60%, whereas idle rate and blocking rate are decreased in the range of and , respectively, as compared with conventional graph coloring and graph‐based dynamic frequency reuse schemes. Copyright © 2013 John Wiley & Sons, Ltd.

On a single machine-scheduling problem with separated position and resource effects

Rudek and Rudek [1] studied a single machine-scheduling problem to minimize the makespan, in which the processing time of each job depends on its position and an amount of a common non-renewable limited resource allocated to it. For this problem, the two effects are separated in the sense that an optimal decision with respect to the job positions can be taken independently of that with respect to the resource allocation. The authors suggested an time algorithm to find an optimal resource allocation. We show that the proof of optimality of this algorithm is incorrect. Further, we demonstrate that the resource allocation problem is equivalent to the well-known fractional knapsack problem, for which there exist and O(n) time algorithms.

Overload-state downlink resource allocation in LTE MAC layer

This paper presents a quality of service-aware resource scheduling algorithm targeted for downlink traffic in the UMTS Long-Term Evolution Medium Access Control layer with emphasis for support in an overload state. Specifically, multiple real-time traffic metrics including latency, loss, and queue depth are normalized and processed via an array of ranking functions and multiplexed into an aggregate rank. This aggregate rank is then used in a fractional knapsack scheduling implementation. Experiments with this scheduling mechanism show improved traffic processing intelligence over existing scheduling models, especially in overload states.

Resource management in software as a service using the knapsack problem model

Abstract This paper proposes a resource allocation model for “Software as a Service” systems that maximizes the service provider's revenues and the resource utilization under a heavy load. Employing the elasticity of virtualized infrastructures, the proposed model dictates that system resources must be fully exploited by incoming jobs, even if they do not satisfy their requirements completely. This yields a higher Service Level Agreement violation probability, which is mitigated by the assignment of more resources when these become available. The problem is deduced to the Fractional Knapsack problem and the heuristic solution is implemented in the frame of a SOA environment.

Learning automata-based solutions to the optimal web polling problem modelled as a nonlinear fractional knapsack problem

We consider the problem of polling web pages as a strategy for monitoring the world wide web. The problem consists of repeatedly polling a selection of web pages so that changes that occur over time are detected. In particular, we consider the case where we are constrained to poll a maximum number of web pages per unit of time, and this constraint is typically dictated by the governing communication bandwidth, and by the speed limitations associated with the processing. Since only a fraction of the web pages can be polled within a given unit of time, the issue at stake is one of determining which web pages are to be polled, and we attempt to do it in a manner that maximizes the number of changes detected. We solve the problem by first modelling it as a stochastic nonlinear fractional knapsack problem. We then present an online learning automata (LA) system, namely, the hierarchy of twofold resource allocation automata (H-TRAA), whose primitive component is a twofold resource allocation automaton (TRAA). Both the TRAA and the H-TRAA have been proven to be asymptotically optimal. Finally, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the learning automata knapsack game (LAKG) which represents the state-of-the-art for this problem. Further, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those which deal with the world wide web.

Video personalization in heterogeneous and resource-constrained environments

Access to multimedia data and multimedia services is becoming increasingly common in networked mobile environments. In such environments, both the mobile client devices and multimedia servers are typically resource constrained. Moreover, the mobile client population is often heterogeneous in terms of the clients' preferences with regard to multimedia content, the clients' quality of service requirements and system-level resource constraints. In order to provide a resource-constrained mobile client with its desired video content, it is often necessary to personalize the requested multimedia content in a manner that satisfies simultaneously the various client-specified content preferences and the system-level resource constraints. Also, in order to simultaneously reduce the client-experienced latency, provide optimal quality of service to the clients and ensure efficient utilization of server and network resources, it is necessary to perform client request aggregation on the server end. To this end, a video personalization strategy is proposed to provide mobile, resource-constrained clients with personalized video content that is most relevant to the clients' requests while satisfying simultaneously multiple client-side system-level resource constraints. A client request aggregation strategy is also proposed to cluster client requests with similar video content preferences and similar client-side resource constraints such that the number of requests the server needs to process and service, and the client-experienced latency are both reduced simultaneously. The primary contributions of the paper are: (1) the formulation and implementation of a Multiple-choice Multi-dimensional Knapsack Problem (MMKP)-based video personalization strategy; and (2) the design and implementation of a client request aggregation strategy based on a multi-stage clustering algorithm. Experimental results comparing the proposed MMKP-based video personalization strategy to existing 0/1 Knapsack Problem (0/1KP)-based and the Fractional Knapsack Problem (FKP)-based video personalization strategies are presented. It is observed that: (1) the proposed MMKP-based personalization strategy includes more relevant video content in response to the client's request compared to the existing 0/1KP-based and FKP-based personalization strategies; and (2) in contrast to the 0/1KP-based and FKP-based personalization strategies which can satisfy only a single client-side resource constraint at a time, the proposed MMKP-based personalization strategy is capable of satisfying multiple client-side resource constraints simultaneously. Experimental results comparing the client-experienced latency with and without the proposed server-side client request aggregation strategy are also presented. It is shown that the proposed client request aggregation strategy reduces the mean client-experienced latency without significant reduction in the average relevance of the delivered video content and without significant deviation in the client-side resources actually consumed by the delivered video content from the client-specified resource constraints.

Dynamic bandwidth provisioning using ARIMA-based traffic forecasting for Mobile WiMAX

With fast proliferation of QoS-enabled wireless packet networks, need for effective QoS control is increasing. In this paper, we focus on QoS provisioning in Mobile WiMAX access service network (ASN). We investigate a dynamic bandwidth provisioning method that can help to increase resource utilization. Our approach consists of two stages: traffic forecasting, followed by bandwidth provisioning. For the first stage, we use auto-regressive integrated moving average (ARIMA) model to forecast traffic based on online measurement. For the second stage, we use a bandwidth provisioning scheme that allocates bandwidths depending on the traffic forecasting. We modeled our problem as a Fractional Knapsack Problem for which we used a greedy algorithm in order to find an approximate solution. Through simulation studies with real-world data sets, we found that our approach could increase the bandwidth for the real-time traffic class and guarantee adequate service quality for the nonreal-time traffic class as well, while maximizing resource utilization.

Optimal sampling for estimation with constrained resources using a learning automaton-based solution for the nonlinear fractional knapsack problem

While training and estimation for Pattern Recognition (PR) have been extensively studied, the question of achieving these when the resources are both limited and constrained is relatively open. This is the focus of this paper. We consider the problem of allocating limited sampling resources in a “real-time” manner, with the explicit purpose of estimating multiple binomial proportions (the extension of these results to non-binomial proportions is, in our opinion, rather straightforward). More specifically, the user is presented with ‘n’ training sets of data points, S 1,S 2,…,S n , where the set S i has N i points drawn from two classes {ω 1,ω 2}. A random sample in set S i belongs to ω 1 with probability u i and to ω 2 with probability 1−u i , with {u i }, i=1,2,…n, being the quantities to be learnt. The problem is both interesting and non-trivial because while both n and each N i are large, the number of samples that can be drawn is bounded by a constant, c. A web-related problem which is based on this model (Snaprud et al., The Accessibility for All Conference, 2003) is intriguing because the sampling resources can only be allocated optimally if the binomial proportions are already known. Further, no non-automaton solution has ever been reported if these proportions are unknown and must be sampled.Using the general LA philosophy as a paradigm to tackle this real-life problem, our scheme improves a current solution in an online manner, through a series of informed guesses which move towards the optimal solution. We solve the problem by first modelling it as a Stochastic Non-linear Fractional Knapsack Problem. We then present a completely new on-line Learning Automata (LA) system, namely, the Hierarchy of Twofold Resource Allocation Automata (H-TRAA), whose primitive component is a Twofold Resource Allocation Automaton (TRAA), both of which are asymptotically optimal. Furthermore, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the Learning Automata Knapsack Game (LAKG) which represents the state-of-the-art. Finally, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those dealing with the world wide web.

Solving Stochastic Nonlinear Resource Allocation Problems Using a Hierarchy of Twofold Resource Allocation Automata

In a multitude of real-world situations, resources must be allocated based on incomplete and noisy information. However, in many cases, incomplete and noisy information render traditional resource allocation techniques ineffective. The decentralized Learning Automata Knapsack Game (LAKG) was recently proposed for solving one such class of problems, namely the class of Stochastic Nonlinear Fractional Knapsack Problems. Empirically, the LAKG was shown to yield a superior performance when compared to methods which are based on traditional parameter estimation schemes. This paper presents a completely new online Learning Automata (LA) system, namely the Hierarchy of Twofold Resource Allocation Automata (H-TRAA). In terms of contributions, we first of all, note that the primitive component of the H-TRAA is a Twofold Resource Allocation Automaton (TRAA) which possesses novelty in the field of LA. Second, the paper contains a formal analysis of the TRAA, including a rigorous proof for its convergence. Third, the paper proves the convergence of the H-TRAA itself. Finally, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the LAKG for simulated data pertaining to two-material unit-value functions. Indeed, in contrast to the LAKG, the H-TRAA scales sublinearly. Consequently, we believe that the H-TRAA opens avenues for handling demanding real-world applications such as the allocation of sampling resources in large-scale Web accessibility assessment problems. We are currently working on applying the H-TRAA solution to the web-polling and sample-size detection problems applicable to the world wide web.

Client-centered multimedia content adaptation

The design and implementation of a client-centered multimedia content adaptation system suitable for a mobile environment comprising of resource-constrained handheld devices or clients is described. The primary contributions of this work are: (1) the overall architecture of the client-centered content adaptation system, (2) a data-driven multi-level Hidden Markov model (HMM)-based approach to perform both video segmentation and video indexing in a single pass, and (3) the formulation and implementation of a Multiple-choice Multidimensional Knapsack Problem (MMKP)-based video personalization strategy. In order to segment and index video data, a video stream is modeled at both the semantic unit level and video program level. These models are learned entirely from training data and no domain-dependent knowledge about the structure of video programs is used. This makes the system capable of handling various kinds of videos without having to manually redefine the program model. The proposed MMKP-based personalization strategy is shown to include more relevant video content in response to the client's request than the existing 0/1 knapsack problem and fractional knapsack problem-based strategies, and is capable of satisfying multiple client-side constraints simultaneously. Experimental results on CNN news videos and Major League Soccer (MLS) videos are presented and analyzed.

A memetic model of evolutionary PSO for computational finance applications

Motivated by the compensatory property of EA and PSO, where the latter can enhance solutions generated from the evolutionary operations by exploiting their individual memory and social knowledge of the swarm, this paper examines the implementation of PSO as a local optimizer for fine tuning in evolutionary search. The proposed approach is evaluated on applications from the field of computational finance, namely portfolio optimization and time series forecasting. Exploiting the structural similarity between these two problems and the non-linear fractional knapsack problem, an instance of the latter is generalized and implemented as the preliminary test platform for the proposed EA–PSO hybrid model. The experimental results demonstrate the positive effects of this memetic synergy and reveal general design guidelines for the implementation of PSO as a local optimizer. Algorithmic performance improvements are similarly evident when extending to the real-world optimization problems under the appropriate integration of PSO with EA.

Learning Automata-Based Solutions to the Nonlinear Fractional Knapsack Problem With Applications to Optimal Resource Allocation

This paper considers the nonlinear fractional knapsack problem and demonstrates how its solution can be effectively applied to two resource allocation problems dealing with the World Wide Web. The novel solution involves a "team" of deterministic learning automata (LA). The first real-life problem relates to resource allocation in web monitoring so as to "optimize" information discovery when the polling capacity is constrained. The disadvantages of the currently reported solutions are explained in this paper. The second problem concerns allocating limited sampling resources in a "real-time" manner with the purpose of estimating multiple binomial proportions. This is the scenario encountered when the user has to evaluate multiple web sites by accessing a limited number of web pages, and the proportions of interest are the fraction of each web site that is successfully validated by an HTML validator. Using the general LA paradigm to tackle both of the real-life problems, the proposed scheme improves a current solution in an online manner through a series of informed guesses that move toward the optimal solution. At the heart of the scheme, a team of deterministic LA performs a controlled random walk on a discretized solution space. Comprehensive experimental results demonstrate that the discretization resolution determines the precision of the scheme, and that for a given precision, the current solution (to both problems) is consistently improved until a nearly optimal solution is found--even for switching environments. Thus, the scheme, while being novel to the entire field of LA, also efficiently handles a class of resource allocation problems previously not addressed in the literature.

Approximation algorithms for fractional knapsack problems

We consider the combinatorial optimization problem where the objective function is the ratio of two linear functions. This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function. We propose for the original problem an approximation scheme based on the one hand upon an approximation scheme for the auxiliary problem and on the other hand upon a constant approximation algorithm for the original problem. As an example of the method we propose an O( n 2) time 1 2 -approximation algorithm for the fractional 0-1 knapsack problem (BFKP) which combined with a known approximation scheme for the 0-1 linear knapsack problem running in O( n 3/ ε) leads to a fully polynomial-time approximation scheme (FPTAS) for BFKP with time complexity O( n 3/ ε). In the same way we propose a FPTAS for the unbounded fractional knapsack problem with time complexity O(n 2+n log(1/ε)+(1/ε) 3) .

Value considerations in three-dimensional packing — A heuristic procedure using the fractional knapsack problem

Not much work has been done on three-dimensional bin packing problem because of the larger number of complexities occurring as compared to packing problems in lower dimensions. Almost all solution procedures developed for bin packing are heuristic in nature. A heuristic algorithm has been presented here for packing a given set of regular sized boxes of different values into a given set of regular containers. The objective is to maximize the total value of the boxes packed into the given containers. The heuristic is based on an Integer Programming formulation that uses the fractional knapsack problem as a column generation subproblem.