Tree Knapsack Problem Research Articles

A novel two-stage approach for energy-efficient timetabling for an urban rail transit network

Urban rail transit (URT) is the backbone transport mode in metropolitan areas to accommodate large travel demands. The high energy consumption of URT becomes a hotspot problem due to the ever-increasing operation mileages and pressing agendas of carbon neutralization. The high model complexity and inconsistency in the objectives of minimizing passenger travel time and operational energy consumption are the main challenges for energy-efficient timetabling for a URT network with multiple interlinked lines. This study proposes a general model framework of timetabling and passenger path choice in a URT network to minimize energy consumption under passenger travel time constraints. To obtain satisfactory energy-efficient nonuniform timetables, we suggest a novel model reformulation as a tree knapsack problem to determine train running times by a pseudo-polynomial dynamic programming algorithm in the first stage. Furthermore, a heuristic sequencing method is developed to determine nonuniform headways and dwell times in the second stage. The suggested model framework and solution algorithm are tested using a real-world URT network, and the results show that energy consumption can be considerably reduced given certain travel time increments.

Exact algorithms for the product configuration problem

AbstractA software product line (SPL) is a software engineering reusability approach used to create different applications from a shared set of common software features. The product configuration problem (PCP) consists in finding the product, among all the products allowed by the SPL, that best fits the customer's preferences and requirements. PCP is a generalization of the tree knapsack problem (TKP) with additional AND, XOR, and OR constraints. Therefore, this problem is also NP‐hard. Previous work in the literature could solve only instances with up to 500 features. In this paper, we propose a pseudo‐polynomial time dynamic programming algorithm that solves PCP in , where n is the number of features and H is the customer's budget. We also propose an integer linear programming formulation for PCP, which is solved by the CPLEX's branch‐and‐bound algorithm. Computational experiments, performed on realistic instances from the literature, showed that the proposed algorithms were able to find optimal solutions for all the instances with up to 10,000 features.

A new opposition crow search optimizer-based two-step approach for controlled intentional islanding in microgrids

Crow search optimizer is considered as the latest meta-heuristic algorithm that is influenced by crow’s behavior. The proposed oppositional crow search optimizer (OCSO) is intended here, for solving the intentional islanding problem. This paper has proposed a novel two-step method by considering the most significant factors such as the constraints of line capacity, bus voltage, load priority, load controllability, problem occurred due to spacing of solutions as well as the capability of integrating the islands to produce higher intentional islands. Initially, the tree knapsack problem is considered as an intentional islanding issue and therefore the OCSO algorithm is employed in solving such shortcomings. In OCSO approach, the opposition-based generation jumping and population initialization concept are used in crow search optimizer for improving the convergence profile and computational speed. In next process, island feasibility is verified by means of conducting power flow computation and providing significant modifications. Six distributed generations containing IEEE 69-bus test system are employed in validating experimentally the efficiency of the proposed approach, and comparison was done for the obtained results with the other existing approaches. The comparative analysis is evaluated to enhance the level of reliability, particularly critical load.

A feasible method for controlled intentional islanding in microgrids based on PSO algorithm

Abstract Intentional islanding is a suitable approach to increase the system reliability in situations where the electrical connection between the microgrid and the upstream network is lost. In this paper, an innovated two-stage method for the intentional islanding process in microgrids is proposed. The important and practical issues such as the load controllability; load priorities; voltage and the line capacity constraints; reduction of problem solution space and ability to make larger islands are taken into account. In the first stage, the problem is relaxed by considering the problem as an optimization problem known as Tree Knapsack Problem (TKP) solved by Particle Swarm Optimization (PSO). In the second stage, the power flow is calculated and the constraints are verified, then adjusting measures will be taken. The proposed method is conducted on the IEEE 69-bus test system with 6 DGs and the results are compared with other methods. Moreover, for real time verification, the obtained results are simulated by DIgSILENT/Power Factory software package. The simulation results suggest that the proposed algorithm is a valid method for the intentional islanding process.

Summarizing a Document by Trimming the Discourse Tree

Recent studies on extractive text summarization formulate it as a combinatorial optimization problem, extracting the optimal subset from a set of the textual units that maximizes an objective function without violating the length constraint. Although these methods successfully improve automatic evaluation scores, they do not consider the discourse structure in the source document. Thus, summaries generated by these methods may lack logical coherence. In previous work, we proposed a method that exploits a discourse tree structure to produce coherent summaries. By transforming a traditional discourse tree, namely a rhetorical structure theory-based discourse tree (RST-DT), into a dependency-based discourse tree (DEP-DT), we formulated the summarization procedure as a Tree Knapsack Problem whose tree corresponds to the DEP-DT. This paper extends the work with a detailed discussion of the approach together with a novel efficient dynamic programming algorithm for solving the Tree Knapsack Problem. Experiments show that our method not only achieved the highest score in both automatic and human evaluation, but also obtained good performance in terms of the linguistic qualities of the summaries.

Island partition of the distribution system with distributed generation

In this paper, a novel optimum island partition model based on Tree Knapsack Problem (TKP) is presented for the distribution system integrated with distributed generation (DG), and a Depth-first Dynamic Programming Algorithm (DPA) is used to solve this model. With the considerations of the load priority, controlled/uncontrolled loads, and the constraints of power balance, voltage and equipment capacity, the model can meet the practical engineering requirements very well. The island partition problem of the distribution system integrated with multiple DGs is first decomposed into multiple TKPs, each of which is solved by DPA respectively. Then, the initial optimum island partition scheme is gained through an island combination procedure, and the final island partition scheme is obtained after feasibility checking and adjustment. Since the algorithm proposed owns the advantages of strong theoretical foundation and low computational complexity, it can find the approximate optimal solution within a limited time. The results of examples demonstrate the validity of the new model and algorithm.

Solving the Extended Tree Knapsack Problem with fixed cost flow expansion functions

Parts of the Local Access Telecommunication Network planning problem may be modelled as an Extended Tree Knapsack Problem. The Local Access Telecommunication Network can contribute up to 60% of the total network costs. This paper presents partitioning algorithms that use standard off-the-shelf software coupled with enhanced modelling. Enhancements to the algorithms and empirical results for both the Tree Knapsack Problem and Extended Tree Knapsack Problem are presented.

Tree knapsack approaches for local access network design

In the process of solving many forms of the local access network design problem, the basic model of the tree knapsack problem (TKP) is used as a building block for the search engine of the solution strategy. Various solution strategies can be used to solve this problem. An approach that use standard software coupled with enhanced modelling is presented for the TKP. Enhanced modelling is used to partition the TKP into sub-problems that is easier to solve using standard off the shelve software. The basic approach is described and empirical work is presented. Empirical comparisons are also given relating this approach with some algorithms suggested by other authors.

Exact and Heuristic Algorithms for Dynamic Tree Simplification

The Tree Knapsack Problem (TKP) is a 0???1 integer programming problem where hierarchy constraints are enforced. If a node is selected for packing into the knapsack, all the ancestor nodes on the path from the root to the selected node are packed as well. One apparent application of this problem is the simplification of computer graphics models. Real applications also use alternative representations of the nodes or whole subtrees, called impostors, to provide simplified trees that are visually acceptable. To account for this simplification, we introduce a generalized TKP, called Exclusive Multiple Choice Tree Knapsack Problem (EMCTKP). We present a dynamic programming algorithm to solve EMCTKP and a heuristic, called Lazy Iterative Arrangement, which reuses previous EMCTKP solutions to solve new instances of the problem. We show that this algorithm and heuristic reduce significantly the computation time of EMCTKP problems when changes in their parameters have spatial and temporal coherence. We also compare our algorithm with commercial integer programming solvers, and show that in our case the computation time grows linearly with the size of the problem tree and the available resources, while for generic IP solvers it is unpredictable and varies over a wide range of values.

The critical-item, upper bounds, and a branch-and-bound algorithm for the tree knapsack problem

The tree knapsack problem (TKP) is a generalized 0–1 knapsack problem where all the items (nodes) are subjected to a partial ordering represented by a rooted tree. If a node is selected to be packed into the knapsack, then all the items on the path from the selected node to the root must also be packed. In this paper, we first define the so-called critical-item for the TKP and present an algorithm to find it. Then, we develop several upper bounds for the TKP. Finally, we present a branch-and-bound procedure for solving the TKP. Our computational results might suggest that our code is currently the most efficient procedure available in the literature. © 1998 John Wiley & Sons, Inc. Networks 31: 205–216, 1998

A Depth-First Dynamic Programming Algorithm for the Tree Knapsack Problem

The Tree Knapsack Problem (TKP) can be regarded as a 0–1 knapsack problem on a rooted tree T such that if a node is selected into a knapsack, then all nodes on the path from the selected node to the root node must also be selected into the knapsack. In this paper, we develop a pseudopolynomial-time algorithm for TKP, the depth-first dynamic programming algorithm. We prove that a TKP can be solved by our algorithm in θ(nH) time, where n is the number of nodes in T and H is the given capacity. We also report the computational results of the depth-first dynamic programming algorithm.

A depth-first dynamic programming procedure for the extended tree knapsack problem in local access network design

The Extended Tree Knapsack Problem (ETKP) is a generalized version of the Tree Knapsack Problem where an arbitrary nonlinear traffic?flow cost is imposed. This problem can be solved by the straight?forward "bottom?up" approach with a time complexity of O(nH 2), where n is the number of nodes in the tree, and H is the knapsack capacity. In this paper, we show that if the traffic?flow cost function is the cable expansion cost, which occurs in the Local Access Telecommunication Network (LATN) expansion, this special ETKP can be solved by a depth?first dynamic programming procedure in a time complexity of O(n?H), where ? is the largest existing cable capacity in LATN. This result indicates that the depth?first dynamic programming algorithm can be applied for solving a general class of tree optimization problems. The computational results of our algorithm for the ETKP are also provided.