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An Efficient greedy Minimum Spanning Tree algorithm Based on Vertex Associative Cycle Detection Method

Procedia Computer Science 2016年 92卷 513-519页

作者： Prantik Biswas Mansi Goel Harshita Negi Megha Datta National Institute of Technology Kurukshetra 136119 India

The minimal spanning tree problem is a popular problem of discrete optimization. Numerous algorithms have been developed using the traditional approach but with the emergence of modern-day complex data structures, new algorithms have been proposed which are more complex yet asymptotically efficient. In this paper we present a cycle detection based greedy algorithm, to obtain a minimal spanning tree of a given input weighted undirected graph. The algorithm operates on the idea that every connected graph without any cycle is a tree. At successive iterations, the algorithm selects and tests if the highest degree vertex is a member of any cycle to remove the most expensive edge from the cycle associated with it. The iteration continues until all the cycles are eliminated to obtain the resultant minimal spanning tree. The simplicity of the algorithm makes it easier to understand and implement in any high-level languages. The proposed approach will be beneficial in analyzing certain class of problems in science and engineering.

关键词： Discrete Optimization Minimal Spanning Tree greedy algorithm Cycle Detection Graph algorithm.

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Approximation algorithm for partial set multicover versus full set multicover

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DISCRETE MATHEMATICS algorithmS AND APPLICATIONS 2018年第2期10卷

作者： Liu, Pengcheng Huang, Xiaohui Zhejiang Normal Univ Coll Math Phys & Informat Engn Jinhua 321004 Zhejiang Peoples R China Zhejiang Normal Univ Lib & Informat Ctr Jinhua 321004 Zhejiang Peoples R China

In a (full) set multicover (SMC) problem, the goal is to select a subcollection of sets to fully cover all elements, where an element is fully covered if it belongs to at least a required number of sets of the selected subcollection. In a partial set multicover (PSMC) problem, it is sufficient to fully cover a required fraction of elements. In this paper, we present a greedy algorithm for PSMC, and analyze the ratio between the cost of the computed solution and the cost of an optimal solution to the corresponding (full) SMC problem. We shall call this ratio as a partial-versus-full ratio. The motivation for such an analysis is to show that PSMC is fairly economic even when it can only be calculated approximately. It turns out that the partial-versus-full ratio of our algorithm is at most ln r/1-rho, where r is the requirement of an element, and rho is the fraction of elements required to be fully covered. An example is given showing that this ratio is tight up to a constant.

关键词： Partial set multicover greedy algorithm performance ratio

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General bounds for incremental maximization 44

General bounds for incremental maximization

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44th International Colloquium on Automata, Languages, and Programming, ICALP 2017

作者： Bernstein, Aaron Disser, Yann Groß, Martin TU Berlin Berlin Germany TU Darmstadt Darmstadt Germany University of Waterloo Waterloo Canada

ISBN: (纸本)9783959770415

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) 2.313 for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems. © Aaron Bernstein, Yann Disser, and Martin Groß;.

关键词： Cardinality constraint Competitive analysis greedy algorithm Incremental optimization Maximization problems

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Theoritical and numerical studies on the graph partitioning problem

Theoritical and numerical studies on the graph partitioning ...

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作者： Haeder Younis Ghawi Althoby Normandie Universite

学位级别：博士

Given G=(V, E) a connected undirected graph and a positive integer β(n), where n is number of vertices, the vertex separator problem (VSP) is to find a partition of V into three classes A, B and C such that there is no edge between A and B, max{|A|,|B|} less than or equal β(n), and |C| is minimum. In this thesis, we consider an integer programming formulation for this problem. We describe some valid inequalties and using these results to develop algorithms based on neighborhood scheme. We also study st-connected vertex separator problem. Let s and t be two disjoint vertices of V, not adjacent. A st-connected separator in the graph G is a subset S of V\{s, t} such that there are no more paths between s and t in G[G\S] and the graph G[S] is connected. The st-connected vertex sperator problem consists in finding such subset with minimum cardinality. We propose three formulations for this problem and give some valid inequalities for the polyhedron associated with this problem. We develop also an efficient heuristic to solve this problem.

关键词： Combinatorial optimization Partitioning graph Heuristics greedy algorithm Branch and bound algorithm

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Efficient and exact mesh deformation using multiscale RBF interpolation 55

Efficient and exact mesh deformation using multiscale RBF in...

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55th AIAA Aerospace Sciences Meeting

作者： Kedward, L. Allen, C.B. Rendall, T.C.S. Department of Aerospace Engineering University of Bristol BristolBS8 1TR United Kingdom

ISBN: (数字)9781624104473

ISBN: (纸本)9781624104473

Mesh deformation schemes are vital in many areas of numerical simulation, and radial basis function(RBF)-based interpolation schemes are popular due to their excellent quality preservation properties. However, the system solution cost scales with moving surface points as N3surface, and so there have been numerous works investigating efficient methods of reducing the data set. However, reduced data methods require the addition of a secondary ‘correction vector field’ to ensure surface points not included in the primary deformation are moved to the correct location, and the volume mesh moved accordingly. A new method is presented here, which captures global and local motions at multiple scales using all the surface points, and so there is no need for a correction stage. The multiscale formulation developed means that although all surface points are used and a single interpolation built, the cost and conditioning issues associated with RBF methods are eliminated while still retaining exact recovery of the surface. Moreover, the sparsity introduced can be exploited, using an existing wall distance function, to further reduce the cost. The method is com- pared with a conventional greedy method on two- and three-dimensional meshes with large deformations. It is shown that mesh quality is always comparable with or better than with the greedy method, and cost and complexity is also comparable or cheaper for all stages. The most expensive stage of reduced point methods is surface mesh preprocessing, and the cost is reduced significantly here;a three or four orders of magnitude reduction in cost is demonstrated compared to greedy-type methods. © 2017 by C.B. Allen.

关键词： Computational Fluid Dynamics Fluid Structure Interaction greedy algorithm Cost Reduction NACA 0012 Aerodynamic Shape Optimization Multivariate Interpolation Structural Element Selection algorithm Mesh Free Methods

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Site selection for small retailers in the food industry using demographic data and network-constrained kernel density estimation

Site selection for small retailers in the food industry usin...

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作者： Bob Houtkooper Wageningen University and Research Center

学位级别：硕士

Despite the high financial risks of site selection, the implementation of spatial data to support site selection for small food retailers is still limited. This thesis proposes a site selection method to help finding multi-location food ventures using a simultaneous approach. The used objective function incorporates variables concerning competition, demand, and site attractiveness. Competition was computed with a network kernel density estimation (KDE). Socio-demographic data, which describe the demand and attractiveness variables, were retrieved and overlaid on a road network. The road segments with the highest utility were found with an optimisation method. In this research, two heuristic optimisation methods were examined: simulated annealing and greedy algorithm. A case which concerned finding the optimal locations of three bakeries in Amersfoort was executed. A user interface was developed to allow the adjustment of weights for the individual variables of the objective function, as well as the number of stores, and the food retailer type. A sensitivity analysis was conducted to assess the sensitivity of the approach to parameter settings. Main findings indicated that the weights for the variables are unknown and differ among food specialists. Owing to the computation load of the network KDE, the method could only be applied at the scale of a city. Also, the available data were not all up-to-date and future demographic predictions were not included. Therefore, the experimental results should be considered as an illustration of the approach. A case was executed for the city of Amersfoort, the Netherlands. The used greedy algorithm was an applicable optimisation method to find single optimal locations. However, the chance that the global optimal was found decreased with increasing numbers of locations. Therefore, when multiple stores locations were to be found, the simulated annealing optimisation method was preferable. Under the default settings, the optimal loca

关键词： site selection small food retail stores network-constrained point patterns spatial optimisation simulated annealing greedy algorithm sensitivity analysis

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Surrogate modeling to enable higher fidelity optimization 18

Surrogate modeling to enable higher fidelity optimization

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18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2017

作者： Vegh, J. Michael Alonso, Juan J. Ilario da Silva, Carlos R. Orra, Tarik H. Stanford University Department of Aeronautics and Astronautics StanfordCA94305 United States EMBRAER São José dos CamposSP12277-901 Brazil

ISBN: (数字)9781624105074

ISBN: (纸本)9781624105074

This paper investigates the integration of surrogate modeling into the design and optimization process of standard and unconventional aircraft. To this end, an open-source design tool called SUAVE, which was developed to allow multifidelity design and optimization of non-conventional aircraft configurations was used. This work identifies some particular design variable choices that allow for smoother, more robust surrogate modeling (as opposed to variables that make the problem highly multimodal). This methodology was applied to two different aircraft cases, consisting of a regional jet aircraft as well as a diesel-electric general aviation aircraft. © 2017 American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.

关键词： Surrogate Model Aircraft Configurations General Aviation Jet Aircraft High Aspect Ratio Aircraft Sizing greedy algorithm Aircraft Design Computational Fluid Dynamics Finite Element Analysis

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greedy adaptive decomposition of signals based on nonlinear Fourier atoms

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INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 2016年第3期14卷

作者： Kou, Kit Ian Li, Hong Univ Macau Fac Sci & Technol Dept Math Taipa Macao Peoples R China Huazhong Univ Sci & Technol Sch Math & Stat Wuhan 430074 Peoples R China

This paper aims at adaptive decomposition of signals in terms of nonlinear Fourier atoms. Each nonlinear Fourier atom is analytic and mono-component. The algorithm is considered as an adaptive greedy procedure based on nonlinear Fourier atoms. The convergence results for the proposed algorithms show that it is suitable to approximate a signal by a linear combinations of nonlinear Fourier atoms. Experiments are presented to illustrate the proposed algorithm and theory.

关键词： greedy algorithm nonlinear Fourier atom analytic signal approximation

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Outlier-Robust greedy Pursuit algorithms in l_p-Space for Sparse Approximation

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2016年第1期64卷 60-75页

作者： Zeng, Wen-Jun So, Hing Cheung Jiang, Xue City Univ Hong Kong Dept Elect Engn Kowloon Hong Kong Peoples R China Arizona State Univ Sch Elect Comp & Energy Engn Tempe AZ 85281 USA

greedy pursuit, which includes matching pursuit (MP) and orthogonal matching pursuit (OMP), is an efficient approach for sparse approximation. However, conventional greedy pursuit algorithms designed for inner product space are not robust against outliers. In this paper, we devise a new definition of correlation in l(p)-space with p > 0, called l(p)-correlation, and introduce the concept of orthogonality in l(p)-space. Based on the l(p)-correlation and l(p)-orthogonality, which are generalizations of the absolute inner product and orthogonality of inner product space, respectively, we develop three greedy pursuit algorithms, namely, l(p)-MP, l(p)-OMP, and weak l(p)-MP, for robust sparse approximation in the presence of outliers. The convergence of the three algorithms is proved. In particular, the l(p)-norm of the residual of each algorithm decays exponentially. It is revealed that the exponential decay factor in the worst case is related to the minimal l(p)-correlation of the residual and the dictionary. Numerical examples on sparse signal recovery and harmonic retrieval in impulsive noise demonstrate that the l(p)-greedy pursuit algorithms with 0 < p < 2 are more outlier-resistant than their counterparts in inner product space.

关键词： Atomic decomposition compressed sensing l(p)-space greedy algorithm matching pursuit orthogonal matching pursuit outlier robust signal recovery sparse approximation

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greedy Hypervolume Subset Selection in Low Dimensions

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EVOLUTIONARY COMPUTATION 2016年第3期24卷 521-544页

作者： Guerreiro, Andreia P. Fonseca, Carlos M. Paquete, Luis Univ Coimbra Dept Informat Engn CISUC Polo 2 P-3030290 Coimbra Portugal

Given a nondominated point set X subset of R-d of size n and a suitable reference point r is an element of R-d, the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size k <= n that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of O(n(k + log n)). In contrast, the best upper bound available for d > 2 is O(n(d/2) log n + n(n-k)). Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of (1 - 1/e) using a greedy strategy. In this article, greedy O(n(k + log n))-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

关键词： Hypervolume indicator multiobjective optimization subset selection monotone submodular function greedy algorithm

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