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Fast Branch and Bound Algorithm for the Travelling Salesman Problem
Abstract
New strategies are proposed for implementing algorithms based on Branch and Bound scheme. Those include two minimal spanning tree lower bound modifications, a design based on the fact that edges in the optimal tour can never cross in the euclidean TSP and parallelization of Branch and Bound scheme. Proposed approaches are compared with primary algorithms.
... The Traveling Salesman Problem (TSP) is a combinatorial optimization problem that has piqued humanity's interest in identifying the shortest distribution route (Dahiya & Sangwan, 2018;Grymin & Jagiełło, 2016;Xiong, Wu, & Wu, 2017). The branch-and-bound algorithm is one of the TSP-solving algorithms that can be used to achieve the goal of this study regarding to the distribution route. ...
... The branch-and-bound algorithm is one of the TSP-solving algorithms that can be used to achieve the goal of this study regarding to the distribution route. It is a recommended method to produce exact algorithms solving NPhard combinatorial optimization and discrete problems (Gmys, Mezmaz, Melab, & Tuyttens, 2016;Grymin & Jagiełło, 2016;Rajarajeswari & Maheswari, 2020;Yu, Xu, & Schober, 2020). The idea behind the algorithm is to use the estimated bound and the least cost search strategy to accelerate the finding of the solution vertex. ...
Article History: The short interval between the collecting and distribution of zakat al-fitr is a recurring issue. As a result, 'amil does not always pay attention to the ideal route, leading in inefficient transportation expenditures. This study aims to minimize the amount of vehicle mileage that affects fuel consumption. The branch-and-bound algorithm was employed to overcome the distribution route optimization problem by proposing the shortest circuit that traverses each district exactly once and returns to its original district. The procedures involve data collecting, graph analysis, branch-and-bound analysis, MATLAB code development, and the recommendation of the best route. The results indicate that the branch-and-bound algorithm can numerically solve the distribution route optimization corresponding to traveling salesman problem. Furthermore, according to a case study of zakat al-fitr distribution conducted by Eradication of Illiteracy Al Quran (PBHA), the total optimal distance of the computational-based algorithm was 152.9 km, with inter-village routes starting from Sidorejo and then via Sumberarum, Pendoworejo, Gerbosari, Banjaroyo, Banjarasri, Sendangagung, Tuksono, Argodadi, Triwidadi, Jatimulyo, Giripurwo, and ends in Sidorejo.
... However, this approach still takes a long time and effort to calculate the final result, especially with large graphs. Typically, when increasing the number of vertices, the processing time can increase exponentially with branch and bound method in [26], or increase with a ratio N 2 2 N with dynamic programming method in [25]. The second one is approximation algorithms being more common for massive graphs, but they may produce poorer quality results than previous algorithms. ...
Deep neural networks (DNNs) have played an increasingly important role in various areas such as computer vision and voice recognition. While training and validation become gradually feasible with high-end general-purpose processors such as graphical processor units (GPU), high throughput inferences in embedded hardware platforms with low hardware resources and power consumption efficiency are still challenging. Binarized neural networks (BNNs) are emerging as a promising method to overcome these challenges by reducing bit widths of DNN data representations with many optimal prior solutions. However, accuracy degradation is a considerable problem of the BNN, compared to the same architecture with full precision, while the binary neural networks still contain significant redundancy for optimization. In this paper, to address the limitations, we implement a streaming accelerator architecture with three optimization techniques: pipelining-unrolling for streaming each layer, weight reuse for parallel computation, and MAC (multiplication-accumulation) compression. Our method first constructs streaming architecture by pipelining-unrolling method to maximize throughput. Next, the weight reuse method with the K-mean cluster is applied to reduce the complexity of the popcount operation. Finally, MAC compression reduces hardware resources used for remaining computation on MAC operations. The implemented hardware accelerator integrated into a state-of-the-art field programable gate array (FPGA) provides the maximum performance of the classification at 1531k frames per second with 98.4% accuracy for the MNIST dataset and 205K frame per second with 80.2% accuracy for the Cifar-10 dataset. Besides, the proposed design’s ratio FPS/LUTs is approximately 57 (MNIST) and 0.707 (Cifar-10), which is much lower than the state-of-the-art design with a comparable throughput and inference accuracy.
Transportation plays a crucial role in both production and service industry. In modern times, the importance of supplying the goods on time to warehouses, production units and finally to the customers is not lost on logistic companies. Vehicle Routing Problems (VRP) models evolved to be more advanced, which led to the growth of computational complexity. Optimizing transportation routes for companies means performing complex computations and doing so in the shortest possible amounts of time. Graphics Processing Units (GPUs) provide massive computation when the needed operations are properly parallelized. nVidia GPUs are equipped with Compute Unified Device Architecture (CUDA), so applying parallel algorithms is not lim- ited to complex workstations or specialized computers. This work emphasizes the value of using parallel Tabu Search (TS) algorithm over sequential TS algorithm and its application to multicriteria discrete optimization of Distance-constrained VRP.
The branch-and-bound (B&B) algorithmic framework has been used successfully to find exact solutions for a wide array of optimization problems. B&B uses a tree search strategy to implicitly enumerate all possible solutions to a given problem, applying pruning rules to eliminate regions of the search space that cannot lead to a better solution. There are three algorithmic components in B&B that can be specified by the user to fine-tune the behavior of the algorithm. These components are the search strategy, the branching strategy, and the pruning rules. This survey presents a description of recent research advances in the design of B&B algorithms, particularly with regards to these three components. Moreover, three future research directions are provided in order to motivate further exploration in these areas.
The Traveling Umpire Problem (TUP) is an optimization problem in which umpires have to be assigned to games in a double round robin tournament. The objective is to obtain a solution with minimum total travel distance over all umpires, while respecting hard constraints on assignments and sequences. Up till now, no general nor dedicated algorithm was able to solve all instances with 12 and 14 teams. We present a novel branch-and-bound approach to the TUP, in which a decomposition scheme coupled with an efficient propagation technique produces the lower bounds. The algorithm is able to generate optimal solutions for all the 12- and 14-team instances as well as for 11 of the 16-team instances. In addition to the new optimal solutions, some new best solutions are presented and other instances have been proven infeasible.
Simple procedures for the traveling salesman problem are presented. The procedures are developed from a new solution generation scheme, which successively exchanges the rows and columns of the matrix of distances. At each stage of the computation, the cost and the tour are uniquely determined.
The traveling salesman problem, TSP for short, has model character in many branches of mathematics, computer science, and operations research. Heuristics, linear programming, and branch and bound, which are still the main components of today's most successful approaches to hard combinatorial optimization problems, were first formulated for the TSP and used to solve practical problem instances. When the theory of NP-completeness developed, the TSP was one of the first problems to be proven NP-hard by Karp in 1972. New algorithmic techniques have first been developed for or at least have been applied to the TSP to show their effectiveness. Examples are branch and bound, Lagrangean relaxation, Lin-Kernighan type methods, simulated annealing, and the field of polyhedral combinatorics for hard combinatorial optimization problems. The chapter presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical instances. The chapter provides guidelines to explain how to attack a TSP instance depending on its size, its structural properties (e.g., metric), the available computation time, and the desired quality of the solution.
We propose a new method of sequential and par-allel speeding up of the process of solving a single machine scheduling problem with total tardiness cost function. This improvement is done on two levels: generating a neigh-borhood inside path relinking method (basing on new approach—blocks in solutions) and parallelization of gener-ating paths. The obtained results are compared to the bench-mark ones taken from the literature. It was possible to find new the best solutions for many benchmark instances by using the proposed method.
The permutation flow shop sequencing problem with the objective of minimizing the sum of the job’s completion times, in literature
known as the F||C
sum
, has been considered. The parallel genetic algorithm based on the island model of migration has been presented. By computer
simulations on Taillard benchmarks [10] and the best known results from literature[9] we have obtained not only acceleration
of the computation’s time but also better quality and stability of the results.
In the late 1950s there was a group of teachers and research assistants at the London School of Economics interested in linear
programming and its extensions, in particular Helen Makower, George Morton, Ailsa Land and Alison Doig. We had considered
the ‘Laundry Van Problem’ until we discovered that it was known as the Traveling Salesman Problem, and had looked at aircraft
timetabling, until quickly realizing that even the planning for the Scottish sector was beyond our capability! Alison Doig
(now Harcourt) had studied the paper trim problem for her Masters project in Melbourne before coming to England.
In this paper we consider two routing problems: traveling salesman problem (TSP, fundamental problem of the combinatorial optimization) and the TSP with times of traveling, times of processing and due dates where the objective is to minimize the total weighted tardiness (TSPTWT). Since problems are NP-hard in the strong sense, we propose a metaheuristic algorithm to determine a good sub-optimal solution.We present an algorithm based on the idea of researching and analyzing local optima.TSP with times of traveling, times of processing and due dates, and with the total weighted tardiness cost function is identical with the single machine total weighted tardiness problem with sequence-dependent setup times. It was possible to find the new best known solutions for 81 of 120 benchmark instances of this scheduling problem using the method proposed here.
New parallel objective function determination methods for the job shop scheduling problem are proposed in this paper, considering makespan and the sum of jobs execution times criteria, however, the methods proposed can be applied also to another popular objective functions such as jobs tardiness or flow time. Parallel Random Access Machine (PRAM) model is applied for the theoretical analysis of algorithm efficiency. The methods need a fine-grained parallelization, therefore the approach proposed is especially devoted to parallel computing systems with fast shared memory (e.g. GPGPU, General-Purpose computing on Graphics Processing Units).
The matter of using scheduling algorithms in parallel computing environments is discussed in this paper. There are proposed methods of parallelizing the criterion function calculations for a single solution and a group of concentrated solutions (local neighborhood) dedicated to being used in metaheuristic approaches. Also a parallel scatter-search metaheuristic is proposed as a multiple-thread approach. Computational experiments are done for the flow shop, the classic NP-hard problem of the combinatorial optimization.
In the classical linear programming problem the behaviour of continuous, nonnegative variables subject to a system of linear inequalities is investigated. One possible generalization of this problem is to relax the continuity condition the variables. This paper presents a simple numerical algorithm for the solution of programming problems in which some or all of the variables can take only discrete values. The algorithm requires no special techniques beyond these used in ordinary linear programming, and lends itself to automatic computing. Its use is illustrated on two numerical examples.