Joint mixability of some integer matrices
Date
2016Source Title
Discrete Optimization
Print ISSN
1572-5286
Publisher
Elsevier B.V. Elsevier BV
Volume
20
Pages
90 - 104
Language
English
Type
ReviewItem 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 permutationJointly 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