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      Joint mixability of some integer matrices

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      Author
      Bellini, F.
      Karaşan, O. E.
      Pınar, M. Ç.
      Date
      2016
      Source Title
      Discrete Optimization
      Print ISSN
      1572-5286
      Publisher
      Elsevier B.V.
       
      Elsevier BV
       
      Volume
      20
      Pages
      90 - 104
      Language
      English
      Type
      Review
      Item Usage Stats
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      Abstract
      We study the problem of permuting each column of a given matrix to achieve minimum maximal row sum or maximum minimal row sum, a problem of interest in probability theory and quantitative finance where quantiles of a random variable expressed as the sum of several random variables with unknown dependence structure are estimated. If the minimum maximal row sum is equal to the maximum minimal row sum the matrix has been termed jointly mixable (see e.g. Haus (2015), Wang and Wang (2015), Wang et al. (2013)). We show that the lack of joint mixability (the joint mixability gap) is not significant, i.e., the gap between the minimum maximal row sum and the maximum minimal row sum is either zero or one for a class of integer matrices including binary and complete consecutive integers matrices. For integer matrices where all entries are drawn from a given set of discrete values, we show that the gap can be as large as the difference between the maximal and minimal elements of the discrete set. The aforementioned result also leads to a polynomial-time approximation algorithm for matrices with restricted domain. Computing the gap for a {0,1,2}-matrix is proved to be equivalent to finding column permutations minimizing the difference between the maximum and minimum row sums. A polynomial procedure for computing the optimum difference by solving the maximum flow problem on an appropriate graph is given. © 2016 Elsevier B.V. All rights reserved.
      Keywords
      Column permutation
      Jointly mixable matrices
      Minimax row sum
      Quantiles
      Approximation algorithms
      Computation theory
      Flow graphs
      Polynomial approximation
      Probability
      Random variables
      Column permutations
      Consecutive integers
      Dependence structures
      Maximum flow problems
      Minimax
      Polynomial time approximation algorithms
      Probability theory
      Quantiles
      Matrix algebra
      Permalink
      http://hdl.handle.net/11693/38124
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.disopt.2016.03.003
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