Greedy algorithm

About: Greedy algorithm is a research topic. Over the lifetime, 15347 publications have been published within this topic receiving 393945 citations.

A man-machine approach toward solving the traveling salesman problem

Abstract: The traveling salesman problem, (form a circuit through N points with no subloops in such a way as to minimize the length of the circuit), is a close kin to many board wiring problems. It has been attacked by many mathematical methods with only meager results. Only for special forms of the problem or for problems with relatively few points can it be solved exactly even using very large amounts of computer time. Heuristic procedures have been proposed and tested with only slightly better results. This paper will describe a computer-aided heuristic technique which uses only a modest amount of computer time in real time to solve large (100-200) point problems. This technique takes advantage of both the computer's and the human's problem solving abilities. The computer is not asked to solve the problem in a brute force way as is the case in many of today's heuristics but it is asked to organize the data for the human in a fashion that allows the human to solve the problem easily.The techniques employed in this paper require that the computer and the human cooperate to find the solution to the problem in reasonable amounts of both of their times. The computer initially uses a series of heuristics that produce groups of points and some partial connections of these points. The human is asked to connect the points within the groups and then connect the groups in a manner that produces a circuit and appears to the human to maximize the ratio of enclosed area to perimeter. The computer takes this solution and uses another set of heuristics to make improvements. The solution is displayed to the human and if he is satisfied the procedure stops; if not the former procedure is repeated until the human is satisfied that cost for finding a better solution exceeds his estimate of the best possible improvement that could be obtained by further work. The heuristic procedures seek to group points around information obtained from solving a series of mathematical programming problems (assignment problems) and some observed correlations between these problems and the traveling salesman problem.The results are very good. The man-machine interaction solution for all problems in the literature is within one per cent of the solution for a fraction of the computer time. The experience to date indicates that the technique can be taught to inexperienced persons and their results, after training, are similar to those of the authors'.

Electric Vehicle Charging Station Placement for Urban Public Bus Systems

Abstract: Due to the low pollution and sustainable properties, using electric buses for public transportation systems has attracted considerable attention, whereas how to recharge the electric buses with long continuous service hours remains an open problem. In this paper, we consider the problem of placing electric vehicle (EV) charging stations at selected bus stops, to minimize the total installation cost of charging stations. Specifically, we study two EV charging station placement cases, with and without considering the limited battery size, which are called ECSP_LB and ECSP problems, respectively. The solution of the ECSP problem achieves the lower bound compared with the solution of the ECSP_LB problem, and the larger the battery size of the EV, the lower the overall cost of the charging station installation. For both cases, we prove that the placement problems under consideration are NP-hard and formulate them into integer linear programming. Specifically, for the ECSP problem we design a linear programming relaxation algorithm to get a suboptimal solution and derive an approximation ratio of the algorithm. Moreover, we derive the condition of the battery size when the ECSP problem can be applied. For the ECSP_LB problem, we show that, for a single bus route, the problem can be optimally solved with a backtracking algorithm, whereas for multiple bus routes we propose two heuristic algorithms, namely, multiple backtracking and greedy algorithms. Finally, simulation results show the effectiveness of the proposed schemes.

Chemical reaction optimization with greedy strategy for the 0-1 knapsack problem

Abstract: The 0-1 knapsack problem (KP01) is a well-known combinatorial optimization problem. It is an NP-hard problem which plays important roles in computing theory and in many real life applications. Chemical reaction optimization (CRO) is a new optimization framework, inspired by the nature of chemical reactions. CRO has demonstrated excellent performance in solving many engineering problems such as the quadratic assignment problem, neural network training, multimodal continuous problems, etc. This paper proposes a new chemical reaction optimization with greedy strategy algorithm (CROG) to solve KP01. The paper also explains the operator design and parameter turning methods for CROG. A new repair function integrating a greedy strategy and random selection is used to repair the infeasible solutions. The experimental results have proven the superior performance of CROG compared to genetic algorithm (GA), ant colony optimization (ACO) and quantum-inspired evolutionary algorithm (QEA).

Greedy local improvement and weighted set packing approximation

Abstract: Given a collection of weighted sets, each containing at most k elements drawn from a finite base set, the k-set packing problem is to find a maximum weight sub-collection of disjoint sets. A greedy algorithm for this problem approximates it to within a factor of k, and a natural local search has been shown to approximate it to within a factor of roughly k?1. However, neither paradigm can yield approximations that improve on this.We present an approximation algorithm for the weighted k-set packing problem that combines the two paradigms by starting with an initial greedy solution and then repeatedly choosing the best possible local improvement. The algorithm has a performance ratio of 2(k+1)/3, which we show is asymptotically tight. This is the first asymptotic improvement over the straightforward ratio of k.

On rectangular partitionings in two dimensions : Algorithms, complexity, and applications

Abstract: Partitioning a multi-dimensional data set into rectangular partitions subject to certain constraints is an important problem that arises in many database applications, including histogram-based selectivity estimation, load-balancing, and construction of index structures. While provably optimal and efficient algorithms exist for partitioning one-dimensional data, the multi-dimensional problem has received less attention, except for a few special cases. As a result, the heuristic partitioning techniques that are used in practice are not well understood, and come with no guarantees on the quality of the solution. In this paper, we present algorithmic and complexity-theoretic results for the fundamental problem of partitioning a two-dimensional array into rectangular tiles of arbitrary size in a way that minimizes the number of tiles required to satisfy a given constraint. Our main results are approximation algorithms for several partitioning problems that provably approximate the optimal solutions within small constant factors, and that run in linear or close to linear time. We also establish the NP-hardness of several partitioning problems, therefore it is unlikely that there are efficient, i.e., polynomial time, algorithms for solving these problems exactly. We also discuss a few applications in which partitioning problems arise. One of the applications is the problem of constructing multi-dimensional histograms. Our results, for example, give an efficient algorithm to construct the V-Optimal histograms which are known to be the most accurate histograms in several selectivity estimation problems. Our algorithms are the first to provide guaranteed bounds on the quality of the solution.