Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center

A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed.     

A Near-Linear Algorithm for the Planar 2-Center Problem

We present an -time algorithm for computing the 2-center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous -time algorithm of [10]. Received May 2, 1995, and in revised form July {8, 1995}.     

关于DNA序列分类问题的模型

本文提出了一种将人工神经元网络用于 DNA分类的方法 .作者首先应用概率统计的方法对 2 0个已知类别的人工 DNA序列进行特征提取 ,形成 DNA序列的特征向量 ,并将之作为样本输入 BP神经网络进行学习 .作者应用了 MATLAB软件包中的 Neural Network Toolbox(神经网络工具箱 )中的反向传播 ( Backpropagation BP)算法来训练神经网络 .在本文中 ,作者构造了两个三层 BP神经网络 ,将提取的 DNA特征向量集作为样本分别输入这两个网络进行学习 .通过训练后 ,将 2 0个未分类的人工序列样本和 1 82个自然序列样本提取特征形成特征向量并输入两个网络进行分类 .结果表明 :本文中提出的分类方法能够以很高的正确率和精度对 DNA序列进行分类 ,将人工神经元网络用于 DNA序列分类是完全可行的     

工序顺序柔性的作业车间调度问题的改进遗传算法求解

针对在工艺设计中提供工序顺序柔性的作业车间调度问题,总结了该问题中柔性工序顺序的类型和特点,并提出了一种求解该问题的改进遗传算法.以尽可能缩短制造周期为目标,结合问题特点,改进了染色体的编码方式,在常用的基于工序顺序的编码方法上融入了基于柔性工序顺序的编码方法,并据此设计了相应的交叉、变异等操作,防止遗传过程中不可行解的产生,避免染色体修复,提高求解效率.最后以MATLAB为工具用某轴承公司的实际生产数据对该算法进行了仿真.通过与不考虑工序顺序柔性的作业车间调度问题遗传算法求解结果进行对比,证明了该算法可行性和有效性.     

A Randomized Homotopy for the Hermitian Eigenpair Problem

Foundations of Computational Mathematics - We describe and analyze a randomized homotopy algorithm for the Hermitian eigenvalue problem. Given an $$ntimes n$$ Hermitian matrix $$A$$ , the...     

A Fast Algorithm for Solving a Simple Search Problem

Computational Mathematics and Mathematical Physics - A fast algorithm for solving the Danskin problem is proposed. The dependence of its solution on parameters is analyzed.     

A Genetic Algorithm for Solving a Capacitated p-Median Problem

Facility-location problems have several applications, such as telecommunications, industrial transportation and distribution. One of the most well-known facility-location problems is the p-median problem. This work addresses an application of the capacitated p-median problem to a real-world problem. We propose a genetic algorithm (GA) to solve the capacitated p-median problem. The proposed GA uses not only conventional genetic operators, but also a new heuristic “hypermutation” operator suggested in this work. The proposed GA is compared with a tabu search algorithm.     

An Approximation Algorithm for a Problem of Partitioning a Sequence into Clusters with Constraints on Their Cardinalities

We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov’s mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal bounds for the Vinogradov mean value, we produce a powerful new kth derivative estimate. Roughly speaking, this improves the van der Corput estimate for k ≥ 4. Various corollaries are given, showing for example that \(\zeta \left( {\sigma + it} \right){ \ll _\varepsilon }{t^{{{\left( {1 - \sigma } \right)}^{3/2}}/2 + \varepsilon }}\) for t ≥ 2 and 0 ≤ σ ≤ 1, for any fixed ε > 0.     

互补问题的一个修改算法

本文给出了一个修改的路径跟踪预测校正非内点算法 ,同时给出了一个新的中心路邻域的表示 .并在此基础上给出了全局和局部收敛性 ,最后给出的数值结果验证了其有效性     

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.     

NP-Hardness of Quadratic Euclidean 1-Mean and 1-Median 2-Clustering Problem with Constraints on the Cluster Sizes

Doklady Mathematics - We consider the problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum (over both clusters) of the intracluster...     

A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time

We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's [3] celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. [6], [7], [9] and to Seidel [20]. As in Chazelle's algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done. Received April 18, 2000, and in revised form December 7, 2000. Online publication June 20, 2001.     

A Multi-Grid Algorithm for Stokes Problem

1.IntroductionConsidertheStokesproblemwhereflisaboundeddomalninRd,d=2or3.Since,withinacodeforthenumer-icalsolutionoftheNavier-Stokesequations,oneneedsanefficientStokessolver,themultigridmethodisveryattractiveforthesolutionofthediscreteanalogueof(1.1).BrezziandDouglas[6Jhaveappliedapenaltyprocedurefor(1.1)withtheC'-piecewiselinearelementofvelocityandpressureandachievedanoptimaJconver-gencerate.Inthispaperweestablishamulti-gridalgorithmforthepenaltyprocedureofStokesproblemandshowthattheconve…     

A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. Received March 6, 1998     

A Pseudo-Polynomial Primal-Dual Algorithm for Globally Solving a Production-Transportation Problem

In this paper, we propose a primal-dual algorithm for solving a class ofproduction-transportation problems. Among m( 2) sources two factoriesexist, which produce given goods at some concave cost and supply them to nterminals. We show that one can globally minimize the total cost ofproduction and transportation by solving a Hitchcock transportation problemwith m sources and n terminals and a minimum linear-cost flow problem withm+n nodes. The number of arithmetic operations required by the algorithm ispseudo-polynomial in the problem input length.     

A Randomized Parallel Algorithm for Planar Graph Isomorphism

We present a parallel randomized algorithm running on aCRCW PRAM, to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms inO(log² n) time withn^{1 + ε}processors, for any ε > 0). Ifnis the number of vertices, our algorithm takesO(log(n)) time with processors and with a probability of failure of 1/nat most. The algorithm needs 2 · log(m) − log(n) + O(log(n)) random bits. The number of random bits can be decreased toO(log(n)) by increasing the number of processors ton^{3/2 + ε}, for any ε > 0. Our parallel algorithm has significantly improved processor efficiency, compared to the previous logarithmic time parallel algorithm of Miller and Reif (Siam J. Comput.20(1991), 1128–1147), which requiresn⁴randomized processors orn⁵deterministic processors.     

A Randomized Parallel Algorithm for Single-Source Shortest Paths

We give a randomized parallel algorithm for computing single-source shortest paths in weighted digraphs. We show that the exact shortest-path problem can be efficiently reduced to solving a series of approximate shortest-path subproblems. Our algorithm for the approximate shortest-path problem is based on the technique used by Ullman and Yannakakis in a parallel algorithm for breadth-first search.     

A Center Cutting Plane Algorithm for a Likelihood Estimate Problem

We describe a method for solving the maximum likelihood estimate problem of a mixing distribution, based on an interior cutting plane algorithm with cuts through analytic centers. From increasingly refined discretized statistical problem models we construct a sequence of inner non-linear problems and solve them approximately applying a primal-dual algorithm to the dual formulation. Refining the statistical problem is equivalent to adding cuts to the inner problems.     

A Modified DIviding RECTangles Algorithm for a Problem in Astrophysics

We present a modification of the DIRECT (DIviding RECTangles) algorithm, called DIRECT-G, to solve a box-constrained global optimization problem arising in the detection of gravitational waves emitted by coalescing binary systems of compact objects. This is a hard problem, since the objective function is highly nonlinear and expensive to evaluate, has a huge number of local extrema and unavailable derivatives. DIRECT performs a sampling of the feasible domain over a set of points that becomes dense in the limit, thus ensuring the everywhere dense convergence; however, it becomes ineffective on significant instances of the problem under consideration, because it tends to produce a uniform coverage of the feasible domain, by oversampling regions that are far from the optimal solution. DIRECT has been modified by embodying information provided by a suitable discretization of the feasible domain, based on the signal theory, which takes into account the variability of the objective function. Numerical experiments show that DIRECT-G largely outperforms DIRECT and the grid search, the latter being the reference algorithm in the astrophysics community. Furthermore, DIRECT-G is comparable with a genetic algorithm specifically developed for the problem. However, DIRECT-G inherits the convergence properties of DIRECT, whereas the genetic algorithm has no guarantee of convergence.     

同顺序流水作业排序问题的一个启发式算法

本文主要给出了同顺序m×n排序问题初始序的选取方法以及通过计算可避免出现高重循环的初始序的排序算法,然后又给出了利用矩阵可行线性质将初始序调试成较优序的可行方法.利用该文方法对n=15,m=3～14的144个例题计算,得出平均相对误差为3.145%的结果,对于m=3与m=4的128个例题计算,得出平均相对误差为0.6306%.统计结果表明该方法可在实际中进行应用.