Multiple-choice Knapsack Problem Research Articles

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.

Chance-Constrained Multiple-Choice Knapsack Problem: Model, Algorithms, and Applications.

The multiple-choice knapsack problem (MCKP) is a classic NP-hard combinatorial optimization problem. Motivated by several significant real-world applications, this work investigates a novel variant of MCKP called the chance-constrained MCKP (CCMCKP), where item weights are random variables. In particular, we focus on the practical scenario of CCMCKP, in which the probability distributions of random weights are unknown and only sample data is available. We first present the problem formulation of CCMCKP and then establish the two benchmark sets. The first set contains synthetic instances, while the second set is designed to simulate a real-world application scenario of a telecommunication company. To solve CCMCKP, we propose a data-driven adaptive local search (DDALS) algorithm. Compared to existing stochastic optimization and distributionally robust optimization methods, the main novelty of DDALS lies in its data-driven solution evaluation approach, which does not make any assumptions about the underlying distributions and is highly effective even when faced with a high intensity of the chance constraint and a limited amount of sample data. Experimental results demonstrate the superiority of DDALS over the baselines on both the benchmarks. Finally, DDALS can serve as the baseline for future research, and the benchmark sets are open-sourced to further promote research on this challenging problem.

Sensitivity analysis of the optimum solution to perturbations of the profit of an item in the multiple-choice knapsack problem

Sensitivity analysis of the optimum solution to perturbations of the profit of an item in the multiple-choice knapsack problem

An exact algorithm for the multiple-choice Knapsack problem with setups

In this paper, an exact method based on Benders’ decomposition is proposed to solve the multiple-choice knapsack problem with setups. The designed method incorporates various techniques to bound the optimal solution effectively. The approach starts by constructing a feasible solution through a series of linear programming relaxations. Each relaxation is augmented with valid constraints to tightly bound both global and local optimal solutions. Subsequently, an initial interval search is established by computing its related lower and upper bounds. These bounds are obtained by solving two tailored linear programming relaxations that account for the cardinality constraint associated with variables representing the families. Once the search interval is determined, it is systematically scanned to find the optimal solution, if one exists. Benders’ decomposition is used for each current sub-interval to bound the local optimal solution. Finally, the proposed method is empirically evaluated on benchmark instances extracted from the literature, where its achieved results are then compared with those obtained by the state-of-the-art Cplex solver. A substantial number of instances have been successfully solved through the proposed method, considering the persistent unresolved state of these instances up to now.

Iterative integer linear programming-based heuristic for solving the multiple-choice knapsack problem with setups

This paper studies the multiple-choice knapsack problem with setup (MCKS) which is a generalization of the knapsack problem with setup (KPS) where the items can be processed in multiple periods. The Integer Linear Programming (ILP) of the MCKS shows its limitations when solved using CPLEX 12.7 solver, and the existing algorithms VNS&IP and VND-LB from the literature outperform the ILP and provide the best-known solutions but solve to optimality only 29 out of 120 instances, respectively. This paper is dedicated to exposing an Iterative Integer linear programming-based heuristic (IILP-H) that deals with the MCKS. The IILP-H has been experimented on the MCKS benchmark instances. A sensitivity analysis of the MCKS parameters is provided. A comparison of the IILP-H with the upper bound obtained by the CPLEX 12.7 solver and the best state-of-the-art algorithms has been conducted. The numerical results show that the IILP-H outperforms all the existing solving techniques when it comes to solution quality and computation time. It reaches 120 out of 120 best solutions, including 77 out of 120 optimal and 69 new best solutions. The complexity of the MCKS and the nature of the solutions obtained by the IILP-H are studied regarding the number of periods to which a family is assigned.

Treatment effect quantiles in stratified randomized experiments and matched observational studies

Summary Evaluating the treatment effect has become an important topic for many applications. However, most existing literature focuses mainly on average treatment effects. When the individual effects are heavy tailed or have outlier values, not only may the average effect not be appropriate for summarizing treatment effects, but also the conventional inference for it can be sensitive and possibly invalid due to poor large-sample approximations. In this paper we focus on quantiles of individual treatment effects, which can be more robust in the presence of extreme individual effects. Moreover, our inference for them is purely randomization based, avoiding any distributional assumptions on the units. We first consider inference in stratified randomized experiments, extending the recent work by Caughey et al. (2021). We show that the computation of valid p-values for testing null hypotheses on quantiles of individual effects can be transformed into instances of the multiple-choice knapsack problem, which can be efficiently solved exactly or slightly conservatively. We then extend our approach to matched observational studies and propose a sensitivity analysis to investigate to what extent our inference on quantiles of individual effects is robust to unmeasured confounding. The proposed randomization inference and sensitivity analysis are simul- taneously valid for all quantiles of individual effects, noting that the analysis for the maximum or minimum individual effect coincides with the conventional analysis assuming constant treatment effects.

Sensitivity Analysis of the Optimum Solution to Perturbations of the Profit of an Item in the Multiple-Choice Knapsack Problem

Sensitivity Analysis of the Optimum Solution to Perturbations of the Profit of an Item in the Multiple-Choice Knapsack Problem

QoE-Driven Adaptive Streaming for Point Clouds

With increasing popularity of virtual reality and augmented reality, application of point clouds is in critical demand as it enables users to freely navigate in an immersive scene with six degrees of freedom. However, point clouds usually comprise large amounts of data, and are thus difficult to stream in bandwidth-constrained networks. It is therefore important, yet challenging, to efficiently stream the resource-intensive point clouds, such that the user's quality of experience (QoE) is guaranteed on a high-level but with a low bandwidth consumption. To this end, we propose a QoE-driven adaptive streaming approach for the tile-based point cloud transmission, to maximize the user's QoE while reducing the transmission redundancy. By exploiting the perspective projection, we specifically model the QoE of a 3D tile as a function of the bitrate of its representation, user's view frustum and spatial position, occlusion between tiles, and the resolution of rendering device. Based on this QoE model, we then formulate the QoE-optimized rate adaptation problem as a multiple-choice knapsack problem, which allocates bitrates for different tiles under a given transmission capacity. It is equivalently converted to a submodular function maximization problem subject to knapsack constraints, and solved by a practical greedy-based algorithm with a theoretical worst-case performance guarantee. The proposed algorithm is able to achieve a near-optimal performance, but with a very low computational complexity. Experimental results further demonstrate superiority of the proposed rate adaptation algorithm over existing schemes, in terms of both user's visual quality and transmission efficiency.

HiTDL: High-Throughput Deep Learning Inference at the Hybrid Mobile Edge

Deep neural networks (DNNs) have become a critical component for inference in modern mobile applications, but the efficient provisioning of DNNs is non-trivial. Existing mobile- and server-based approaches compromise either the inference accuracy or latency. Instead, a hybrid approach can reap the benefits of the two by splitting the DNN at an appropriate layer and running the two parts separately on the mobile and the server respectively. Nevertheless, the DNN throughput in the hybrid approach has not been carefully examined, which is particularly important for edge servers where limited compute resources are shared among multiple DNNs. This article presents HiTDL, a runtime framework for managing multiple DNNs provisioned following the hybrid approach at the edge. HiTDL's mission is to improve edge resource efficiency by optimizing the combined throughput of all co-located DNNs, while still guaranteeing their SLAs. To this end, HiTDL first builds comprehensive performance models for DNN inference latency and throughout with respect to multiple factors including resource availability, DNN partition plan, and cross-DNN interference. HiTDL then uses these models to generate a set of candidate partition plans with SLA guarantees for each DNN. Finally, HiTDL makes global throughput-optimal resource allocation decisions by selecting partition plans from the candidate set for each DNN via solving a fairness-aware multiple-choice knapsack problem. Experimental results based on a prototype implementation show that HiTDL improves the overall throughput of the edge by <inline-formula><tex-math notation="LaTeX">$4.3\times$</tex-math></inline-formula> compared with the state-of-the-art.

Cost-Efficient Workflow Scheduling Algorithm for Applications With Deadline Constraint on Heterogeneous Clouds

In recent years, more and more large-scale data processing and computing workflow applications run on heterogeneous clouds. Such cloud applications with precedence-constrained tasks are usually deadline-constrained and their scheduling is an essential problem faced by cloud providers. Moreover, minimizing the workflow execution cost based on cloud billing periods is also a complex and challenging problem for clouds. In realizing this, we first model the workflow applications as I/O Data-aware Directed Acyclic Graph (DDAG), according to clouds with global storage systems. Then, we mathematically state this deadline-constrained workflow scheduling problem with the goal of minimum execution financial cost. We also prove that the time complexity of this problem is NP-hard by deducing from a multidimensional multiple-choice knapsack problem. Third, we propose a heuristic cost-efficient task scheduling strategy called CETSS, which includes workflow DDAG model building, task subdeadline initialization, greedy workflow scheduling algorithm, and task adjusting method. The greedy workflow scheduling algorithm mainly consists of dynamical task renting billing period sharing method and unscheduled task subdeadline relax technique. We perform rigorous simulations on some synthetic randomly generated applications and real-world applications, such as Epigenomics, CyberShake, and LIGO. The experimental results clearly demonstrate that our proposed heuristic CETSS outperforms the existing algorithms and can effective save the total workflow execution cost. In particular, CETSS is very suitable for large workflow applications.

Online joint bid/daily budget optimization of Internet advertising campaigns

Pay-per-click advertising includes various formats (e.g., search, contextual, social) with a total investment of more than 200 billion USD per year worldwide. An advertiser is given a daily budget to allocate over several campaigns, mainly distinguishing for the ad, target, or channel. Furthermore, publishers choose the ads to display and how to allocate them employing auctioning mechanisms, in which, every day and for each campaign, the advertisers set a bid corresponding to the maximum amount of money per click they are willing to pay and the fraction of the daily budget to invest. In this paper, we study the problem of automating the online joint bid/daily budget optimization of pay-per-click advertising campaigns over multiple channels, and we face the challenging goal of designing techniques with theoretical guarantees that can be applied in real-world applications, where, commonly, data scarcity is a crucial issue. We formulate our problem as a combinatorial semi-bandit problem, which requires solving a special case of the Multiple-Choice Knapsack problem every day. Furthermore, we address data scarcity by designing a model for the dependency of the number of clicks on the bid and daily budget, requiring few parameters at the cost of mild regularity assumptions. We propose two algorithms—the first is randomized, while the second is deterministic—and show that they suffer from a regret that is upper bounded with high probability as O˜(T), where T is the time horizon of the learning process. We experimentally evaluate our algorithms with synthetic settings generated from real data provided by Yahoo!, and we present the results of adopting our algorithms in a real-world application with a daily spent of 1,000 Euros for more than one year.

Bandit Learning-Based Edge Caching for 360-Degree Video Streaming With Switching Cost

Virtual Reality (VR) and Augmented Reality (AR) applications are expected to be a key driver for the 5G network. The need of the MEC support for caching and transcoding of omnidirectional content has been identified as a major topic of research. In this paper, we study the optimal caching problem in an MEC-enabled VR system for tile-based viewport-adaptive 360-degree video streaming. For content caching, the replacement of the cached content inevitably incurs content switching cost, which is largely ignored in caching policy design. In this work, we aim to find the optimal caching policy to optimize the long-term transmission quality by considering both transmission latency and content switching cost. We apply the combinatorial multi-armed bandit (CMAB) theory to solve the problem with no a-priori knowledge on content popularity. Moreover, the CUCB with switching cost (CUCBSC) algorithm is adopted in this scenario. A transformation mechanism is designed to transform the generated single period optimization (SPO) problem into a multiple choice knapsack problem (MCKP). Rigorous theoretical analysis on the performance of the proposed algorithm is also provided. Finally, the effectiveness of the proposed learning based caching policy is confirmed by simulation results in terms of learning, hit-ratio, transmission and content switching delay.respondence) papers with a suitable choice of class options.

Distributed Clustering-Based Cooperative Vehicular Edge Computing for Real-Time Offloading Requests

Mobile vehicles have been considered as potential edge servers to provide computation resources for the emerging Intelligent Transportation System (ITS) applications. However, how to fully utilize the mobile computation resources to satisfy the real-time arrived computation requests has not been explored yet. This work will address the critical challenges of limited computation resources, stringent computation delay and unknown requirement statistics of real-time tasks in realistic vehicular edge computing scenarios. Specifically, we design a distributed clustering strategy to classify vehicles into multiple cooperative edge servers according to the available computation resources, effective connection time and the distribution of tasks&#x2019; expected deadlines. Then, a &#x2018;Less than or Equal to&#x2019; Generalized Assignment Problem (i.e., LEGAP) is formulated to maximize the system service revenue, and on this basis, we propose an offline Bound-and-Bound based Optimal (BBO) algorithm to make periodical scheduling with a global view of tasks&#x2019; requirement statistics. The quick branching is conducted by following a greedy solution and the upper bound at each branch is derived by solving a multiple-choice knapsack problem. In addition, we present an online heuristic algorithm which makes real-time offloading decision with the guarantees that the resource capacities of all the computing servers are never exceeded with new tasks arriving. Through comparing with the other four online algorithms, the BBO algorithm achieves the highest service revenues by offloading tasks with shorter delays, and the online heuristic algorithm has the best performance in improving the service ratios.

A two-phase kernel search variant for the multidimensional multiple-choice knapsack problem

The Multidimensional Multiple-choice Knapsack Problem (MMKP) is a complex combinatorial optimization problem for which finding high quality feasible solutions is very challenging. Recently, several heuristic approaches and a few exact algorithms have been proposed for its solution. These methods have been able to provide new best-known values for benchmark instances although many of them still remain unclosed to optimality. In this paper, we provide a new variant of the heuristic framework Kernel Search and apply it to the MMKP. The proposed variant keeps the method’s main idea of solving a sequence of restricted mixed-integer subproblems but innovates by partitioning the solution process into two different phases with complementary goals. The first phase strives for feasibility and collects important information to dynamically adapt subproblems’ dimension and solution time in the second phase that is focused on getting high quality solutions. This makes the global approach more scalable and efficient.Computational results on different sets of benchmark instances demonstrate that our method is extremely effective outperforming all state-of-the-art heuristics for the MMKP. The method compares extremely well also with respect to exact approaches running for five hours. The proposed algorithm is able to improve the best-known value of 185 out of 276 open benchmark instances and gets 8 out of 17 optimal solutions for the closed ones. The average deviation from optimality is always negligible.

A new combinatorial branch-and-bound algorithm for the Knapsack Problem with Conflicts

We study the Knapsack Problem with Conflicts, a generalization of the Knapsack Problem in which a set of conflicts specifies pairs of items which cannot be simultaneously selected. In this work, we propose a novel combinatorial branch-and-bound algorithm for this problem based on an n-ary branching scheme. Our algorithm effectively combines different procedures for pruning the branch-and-bound nodes based on different relaxations of the Knapsack Problem with Conflicts. Its main elements of novelty are: (i) the adoption of the branching-and-pruned set branching scheme which, while extensively used in the maximum-clique literature, was never successfully employed for solving the Knapsack Problem with Conflicts; (ii) the adoption of the Multiple-Choice Knapsack Problem for the derivation of upper bounds used for pruning the branch-and-bound tree nodes; and (iii) the design of a new upper bound for the latter problem which can be computed very efficiently. Key to our algorithm is its high pruning potential and the low computational effort that it requires to process each branch-and-bound node. An extensive set of experiments carried out on the benchmark instances typically used in the literature shows that, for edge densities ranging from 0.1 to 0.9, our algorithm is faster by up to two orders of magnitude than the state-of-the-art method and by up to several orders of magnitude than a state-of-the-art mixed-integer linear programming solver.

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems. The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of items

A Core-Based Exact Algorithm for the Multidimensional Multiple Choice Knapsack Problem

In the multidimensional multiple choice knapsack problem (MMKP), items with nonnegative profits are partitioned into groups. Each item consumes a predefined nonnegative amount of a set of resources...

Cache-Based Multi-Query Optimization for Data-Intensive Scalable Computing Frameworks

In modern large-scale distributed systems, analytics jobs submitted by various users often share similar work, for example scanning and processing the same subset of data. Instead of optimizing jobs independently, which may result in redundant and wasteful processing, multi-query optimization techniques can be employed to save a considerable amount of cluster resources. In this work, we introduce a novel method combining in-memory cache primitives and multi-query optimization, to improve the efficiency of data-intensive, scalable computing frameworks. By careful selection and exploitation of common (sub)expressions, while satisfying memory constraints, our method transforms a batch of queries into a new, more efficient one which avoids unnecessary recomputations. To find feasible and efficient execution plans, our method uses a cost-based optimization formulation akin to the multiple-choice knapsack problem. Extensive experiments on a prototype implementation of our system show significant benefits of worksharing for both TPC-DS workloads and detailed micro-benchmarks.

Joint optimization of building-envelope and heating-system retrofits at territory scale to enhance decision-aiding

Reduction of energy consumption in the building sector has been identified as a major instrument to tackle global climate change and improve sustainability. In this paper, we propose a methodology to address a retrofit planning problem at a community level, with a building resolution. The resulting tool helps local decision-makers identify pertinent actions to improve the environmental behavior of their territories. Two building retrofit levers are considered, namely envelope insulation and heating systems replacement. Retrofit planning is treated here as a single-objective optimization problem aimed at reducing the total costs of retrofit actions by minimizing their net present value. A multidimensional multiple-choice knapsack problem formulation is proposed through the adoption of adequate decision variables. It suitably balances the complexity induced by the large number of potential retrofit action combinations and the number of variables in the problem and permits a linear formulation. An optimization of virtual building stocks is performed to highlight the developed model’s capacity to tackle large problems (5,000 buildings) in a few minutes. Finally, three analyses finally are led on a real case-study territory, featuring both appropriate retrofit solutions and building stock information. Long-term evaluation of retrofit strategies over the short-term results in an additional 10% reduction of energy consumption and greenhouse gases emissions and encourages thermal insulation. When targeting a 40% reduction in energy demand, retrofit costs ranging from 20 to 800€/m2 are observed. Finally, the developed method was used to draw a CO2 abatement cost curve at territory level. A 70% reduction of emissions can be achieved with costs under 50 €/tCO2e.

Energy-Aware Design of Stochastic Applications With Statistical Deadline and Reliability Guarantees

Energy efficiency, reliability, and real-time are three key requirements of mission-critical embedded systems. Existing approaches over emphasize the worst case design of real-time embedded systems, which will lead to serious waste of resources. In this paper, we aim at the energy-efficient design of soft real-time and reliable applications on uniprocessor embedded systems. We consider soft real-time tasks with stochastic execution durations regarding certain distributions. Thereby, we provide real-time guarantee with probability consideration. We utilize dynamic voltage and frequency scaling (DVFS) for saving energy, and also take into account the impact of DVFS on reliability. Our objective is to minimize the expected energy consumption of the system subject to statistical reliability and deadline constraints. The design optimization problem is a typical multidimensional multiple-choice knapsack problem, which is NP-hard. We first propose a dynamic programming-based optimal algorithm to solve the problem. To reduce the time complexity, we then develop a ( $1+{\beta }$ )-approximation algorithm based on a binary search approach, where ${\beta }$ is the approximating factor. The approximation algorithm can obtain the near-optimal solution with at most ( $1{+\beta }$ ) times of optimal energy cost under given real-time and reliability constraints and has fully polynomial time complexity. Extensive experiments and a real-life synthetic application are conducted to evaluate the performance of the proposed techniques. Compared with existing approaches, the approximation approach can save much energy with low time overhead while guaranteeing the statistical deadline and reliability constraints.