1,537 research outputs found

    A Heuristic Approach to the Consecutive Ones Submatrix Problem

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    أعطيت مصفوفة (0،1)، تم اقتراح مسألة المصفوفة الجزئية ذات الواحدات المتعاقبة والتي تهدف إلى إيجاد تبديل للأعمدة التي تزيد من عدد الأعمدة التي تحتوي معًا على قالب واحد فقط من الواحدات المتعاقبة في كل صف. سيتم اقتراح اسلوب الاستدلال لحل المسألة. كما سيتم دراسة مسألة تقليل القوالب المتتالية ذات الصلة بمسألة المصفوفة الجزئية ذات الواحدات المتعاقبة. تم اقتراح اجراء جديد لتحسين طريقة إدراج العمود. يتم بعد ذلك تقييم مصفوفات العالم الحقيقي ومصفوفات متولدة عشوائيًا من مسألة غطاء المجموعة و تعرض النتائج الحسابية.Given a matrix, the Consecutive Ones Submatrix (C1S) problem which aims to find the permutation of columns that maximizes the number of columns having together only one block of consecutive ones in each row is considered here. A heuristic approach will be suggested to solve the problem. Also, the Consecutive Blocks Minimization (CBM) problem which is related to the consecutive ones submatrix will be considered. The new procedure is proposed to improve the column insertion approach. Then real world and random matrices from the set covering problem will be evaluated and computational results will be highlighted

    Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs

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    Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any binary matrix that does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th International Workshop on Graph-Theoretic Concepts in Computer Scienc

    Computational barriers in minimax submatrix detection

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    This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: when the size of the large matrix pp\to\infty, if the submatrix size k=Θ(pα)k=\Theta(p^{\alpha}) for any α(0,2/3)\alpha\in(0,{2}/{3}), computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial-time test is minimax suboptimal by a polynomial factor in pp; if k=Θ(pα)k=\Theta(p^{\alpha}) for any α(2/3,1)\alpha\in({2}/{3},1), minimax optimal detection can be attained within constant factors in linear time. Using Schatten norm loss as a representative example, we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function. Implications on the hardness of support recovery are also obtained.Comment: Published at http://dx.doi.org/10.1214/14-AOS1300 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm

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    This paper studies the long-existing idea of adding a nice smooth function to "smooth" a non-differentiable objective function in the context of sparse optimization, in particular, the minimization of x1+1/(2α)x22||x||_1+1/(2\alpha)||x||_2^2, where xx is a vector, as well as the minimization of X+1/(2α)XF2||X||_*+1/(2\alpha)||X||_F^2, where XX is a matrix and X||X||_* and XF||X||_F are the nuclear and Frobenius norms of XX, respectively. We show that they can efficiently recover sparse vectors and low-rank matrices. In particular, they enjoy exact and stable recovery guarantees similar to those known for minimizing x1||x||_1 and X||X||_* under the conditions on the sensing operator such as its null-space property, restricted isometry property, spherical section property, or RIPless property. To recover a (nearly) sparse vector x0x^0, minimizing x1+1/(2α)x22||x||_1+1/(2\alpha)||x||_2^2 returns (nearly) the same solution as minimizing x1||x||_1 almost whenever α10x0\alpha\ge 10||x^0||_\infty. The same relation also holds between minimizing X+1/(2α)XF2||X||_*+1/(2\alpha)||X||_F^2 and minimizing X||X||_* for recovering a (nearly) low-rank matrix X0X^0, if α10X02\alpha\ge 10||X^0||_2. Furthermore, we show that the linearized Bregman algorithm for minimizing x1+1/(2α)x22||x||_1+1/(2\alpha)||x||_2^2 subject to Ax=bAx=b enjoys global linear convergence as long as a nonzero solution exists, and we give an explicit rate of convergence. The convergence property does not require a solution solution or any properties on AA. To our knowledge, this is the best known global convergence result for first-order sparse optimization algorithms.Comment: arXiv admin note: text overlap with arXiv:1207.5326 by other author

    Minimising the number of gap-zeros in binary matrices

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    We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows
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