## Joint mixability of some integer matrices

##### 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 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