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Parameterized algorithms for min-max 2-cluster editing

JOURNAL OF COMBINATORIAL OPTIMIZATION 2017年第1期34卷 47-63页

作者： Chen, Li-Hsuan Wu, Bang Ye Natl Chung Cheng Univ Chiayi 621 Taiwan

For a given graph and an integer t, the Min-Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t. It has been shown that the problem can be solved in polynomial time for , where n is the number of vertices. In this paper, we design parameterized algorithms for different ranges of t. Let . We show that the problem is polynomial-time solvable when roughly . When , we design a randomized and a deterministic algorithm with sub-exponential time parameterized complexity, i.e., the problem is in SUBEPT. We also show that the problem can be solved in time for and in time for , where .

关键词： Parameterized algorithm Parameterized complexity Graph modification Clustering randomized algorithm Subexponential algorithm

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Stochastic sampling for deterministic structural topology optimization with many load cases: Density-based and ground structure approaches

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2017年 325卷 463-487页

作者： Zhang, Xiaojia Shelly de Sturler, Eric Paulino, Glaucio H. Georgia Inst Technol Sch Civil & Environm Engn 790 Atlantic Dr Atlanta GA 30332 USA Virginia Tech Dept Math McBryde Hall225 Stanger St Blacksburg VA 24061 USA

We propose an efficient probabilistic method to solve a fully deterministic problem - we present a randomized optimization approach that drastically reduces the enormous computational cost of optimizing designs under many load cases for both continuum and truss topology optimization. Practical structural designs by deterministic topology optimization typically involve many load cases, possibly hundreds or more. The optimal design minimizes a, possibly weighted, average of the compliance under each load case (or some other objective). This means that, in each optimization step, a large finite element problem must be solved for each load case, leading to an enormous computational effort. On the contrary, the proposed randomized optimization method with stochastic sampling requires the solution of only a few (e.g., 5 or 6) finite element problems (large linear systems) per optimization step. Based on simulated annealing, we introduce a damping scheme for the randomized approach. Through numerical examples in two and three dimensions, we demonstrate that the randomization algorithm drastically reduces computational cost to obtain similar final topologies and results (e.g., compliance) to those of standard algorithms. The results indicate that the damping scheme is effective and leads to rapid convergence of the proposed algorithm. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Topology optimization with many load cases Stochastic sampling randomized algorithm Trace estimator Density-based method Ground structure method

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HOUSEHOLDER QR FACTORIZATION WITH RANDOMIZATION FOR COLUMN PIVOTING (HQRRP)

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2017年第2期39卷 C96-C115页

作者： Martinsson, Per-Gunnar Orti, Gregorio Quintana Heavner, Nathan van de Geijn, Robert Univ Colorado Boulder Dept Appl Math Boulder CO 80309 USA Univ Jaume 1 Dept Ingn & Ciencia Comp Castellon de La Plana 12071 Spain Univ Texas Austin Dept Comp Sci Austin TX 78712 USA Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA

A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix matrix multiplications, which greatly limits the benefits that can be derived from so-called blocking of algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic flop count for the proposed method is 2mn(2) (2/3)n(3) for an m x n matrix, identical to that of the best classical unblocked Householder QR factorization algorithm (with or without pivoting). Experiments demonstrate acceleration in speed of close to an order of magnitude relative to the GEQP3 function in LAPACK, when executed on a modern CPU with multiple cores. Further, experiments demonstrate that the quality of the randomized pivot selection strategy is roughly the same as that of classical column pivoting. The described algorithm is made available under open source license and can be used with LAPACK or libflame.

关键词： QR factorization Householder QR pivoting strategy randomized algorithm blocked algorithm

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INTERPOLATIVE BUTTERFLY FACTORIZATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2017年第2期39卷 A503-A531页

作者： Li, Yingzhou Yang, Haizhao Stanford Univ ICME Stanford CA 94305 USA Duke Univ Dept Math Durham NC 27708 USA

This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low-rank property. A preliminary interpolative butterfly factorization is constructed based on interpolative low-rank approximations of the complementary low-rank matrix. A novel sweeping matrix compression technique further compresses the preliminary interpolative butterfly factorization via a sequence of structure-preserving low-rank approximations. The sweeping procedure propagates the low-rank property among neighboring matrix factors to compress dense submatrices in the preliminary butterfly factorization to obtain an optimal one in the butterfly scheme. For an N x N matrix, it takes O(N log N) operations and complexity to construct the factorization as a product of O(log N) sparse matrices, each with O(N) nonzero entries. Hence, it can be applied rapidly in O(N log N) operations. Numerical results are provided to demonstrate the effectiveness of this algorithm.

关键词： data-sparse matrix butterfly algorithm randomized algorithm matrix factorization operator compression nonuniform Fourier transform Fourier integral operators

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An HSS Matrix-Inspired Butterfly-Based Direct Solver for Analyzing Scattering From Two-Dimensional Objects

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 2017年 16卷 1179-1183页

作者： Liu, Yang Guo, Han Michielssen, Eric Univ Michigan Dept Elect Engn & Comp Sci Ann Arbor MI 48109 USA

A butterfly-based fast direct integral equation solver for analyzing high-frequency scattering from two-dimensional objects is presented. The solver leverages a randomized butterfly scheme to compress blocks corresponding to near-and far-field interactions in the discretized forward and inverse electric field integral operators. The observed memory requirements and computational cost of the proposed solver scale as O(Nlog(2)N) and O(N-1.5 logN), respectively. The solver is applied to the analysis of scattering from electrically large objects spanning over 10 000 wavelengths and modeled in terms of five million unknowns.

关键词： Butterfly scheme electromagnetic scattering fast direct solver multilevel matrix decomposition algorithm (MLMDA) randomized algorithm

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Low-rank approximation pursuit for matrix completion

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2017年 95卷 77-89页

作者： Xu, An-Bao Xie, Dongxiu Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Beijing Informat Sci & Technol Univ Sch Sci Beijing 100192 Peoples R China

We consider the matrix completion problem that aims to construct a low rank matrix X that approximates a given large matrix Y from partially known sample data in Y. In this paper we introduce an efficient greedy algorithm for such matrix completions. The greedy algorithm generalizes the orthogonal rank-one matrix pursuit method (OR1MP) by creating s >= 1 candidates per iteration by low-rank matrix approximation. Due to selecting s >= 1 candidates in each iteration step, our approach uses fewer iterations than OR1MP to achieve the same results. Our algorithm is a randomized low-rank approximation method which makes it computationally inexpensive. The algorithm comes in two forms, the standard one which uses the Lanzcos algorithm to find partial SVDs, and another that uses a randomized approach for this part of its work. The storage complexity of this algorithm can be reduced by using an weight updating rule as an economic version algorithm. We prove that all our algorithms are linearly convergent. Numerical experiments on image reconstruction and recommendation problems are included that illustrate the accuracy and efficiency of our algorithms. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Low rank approximation Matrix completion randomized algorithm Rank minimization Matching pursuit

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A POWER-OF-TWO-CHOICES UNBALANCED ALLOCATION PROCESS

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2017年第1期31卷 477-488页

作者： Redlich, Amanda Bowdoin Coll Dept Math Brunswick ME 04011 USA

The well-studied "power of two choices" family of algorithms creates balanced allocations of m balls into n bins by, for each ball, selecting a few bins at random and then placing the item in the least-loaded bin. A natural variation is to create an unbalanced allocation by, for each ball, selecting a few bins at random and then placing the ball in the most-loaded bin. Surprisingly, this variation has not been previously studied. This paper introduces this family of unbalanced allocation processes and begins its analysis. The behavior of the bounded m case is analyzed in detail via differential equations and coupling, and some preliminary results for the general case are presented.

关键词： Achlioptas process differential equations method power of two choices preferential attachment model randomized algorithm

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Narrow sieves for parameterized paths and packings

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JOURNAL OF COMPUTER AND SYSTEM SCIENCES 2017年 87卷 119-139页

作者： Bjorklund, Andreas Husfeldt, Thore Kaski, Petteri Koivisto, Mikko Lund Univ Dept Comp Sci POB 118 SE-22100 Lund Sweden IT Univ Copenhagen DK-2300 Copenhagen Denmark Aalto Univ Dept Informat & Comp Sci HIIT POB 15400 FI-00076 Aalto Finland Univ Helsinki HIIT Dept Comp Sci POB 68 FI-00014 Helsinki Finland

We present parameterized algorithms for the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space. The constant bases of the exponentials are significantly smaller than in previous works;for example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with d colors in time within a polynomial factor of 2((d-1)n/2). Our techniques generalize an algebraic approach studied in various recent works. (c) 2017 Elsevier Inc. All rights reserved.

关键词： Determinant Edge coloring Graph algorithm k-Path Multidimensional matching Sieve Set packing Polynomial identity testing randomized algorithm

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Preconditioned Data Sparsification for Big Data With Applications to PCA and K-Means

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IEEE TRANSACTIONS ON INFORMATION THEORY 2017年第5期63卷 2954-2974页

作者： Pourkamali-Anaraki, Farhad Becker, Stephen Univ Colorado Dept Elect Comp & Energy Engn Boulder CO 80309 USA Univ Colorado Dept Appl Math Boulder CO 80309 USA

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix, and therefore, subsequent processing, such as principal component analysis (PCA) or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.

关键词： Unsupervised learning PCA clustering K-means randomized algorithm big data

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ACCURATE LOW-RANK APPROXIMATIONS VIA A FEW ITERATIONS OF ALTERNATING LEAST SQUARES

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2017年第2期38卷 425-433页

作者： Szlam, Arthur Tulloch, Andrew Tygert, Mark Facebook Artificial Intelligence Res Menlo Pk CA 94025 USA

A few iterations of alternating least squares with a random starting point provably suffice to produce nearly optimal spectral- and Frobenius-norm accuracies of low-rank approximations to a matrix;iterating to convergence of the matrix entries is unnecessary. Such good accuracy is in fact well known for the low-rank approximations calculated via subspace iterations and other well-known methods that happen to produce mathematically the same low-rank approximations as alternating least squares, at least when starting all the methods with the same appropriately random initializations. Thus, software implementing alternating least squares can be retrofitted via appropriate setting of parameters to calculate nearly optimally accurate low-rank approximations highly efficiently, with no need for convergence of the matrix entries. (Even so, convergence could still be helpful for some applications, say to ensure that the approximations are strongly rank-revealing.)

关键词： low-rank approximation principal component analysis alternating least squares alternating minimization randomized algorithm

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