Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems

TL;DR

This paper investigates the problem of matrix complete mixability and related optimization problems, establishing complexity results, approximation algorithms, and polynomial-time solutions under specific conditions, with applications to dependence modeling and quantile estimation.

Contribution

It introduces new complexity results, approximation algorithms, and polynomial-time solutions for complete mixability and related multidimensional assignment problems.

Findings

Deciding complete mixability is NP-complete.

A polynomial 2-approximation algorithm exists for 3-column matrices.

A PTAS is available for fixed number of columns.

Abstract

We call a matrix completely mixable if the entries in its columns can be permuted so that all row sums are equal. If it is not completely mixable, we want to determine the smallest maximal and largest minimal row sum attainable. These values provide a discrete approximation of of minimum variance problems for discrete distributions, a problem motivated by the question how to estimate the $\alpha$-quantile of an aggregate random variable with unknown dependence structure given the marginals of the constituent random variables. We relate this problem to the multidimensional bottleneck assignment problem and show that there exists a polynomial $2$-approximation algorithm if the matrix has only $3$ columns. In general, deciding complete mixability is $\mathcal{NP}$-complete. In particular the swapping algorithm of Puccetti et al. is not an exact method unless $\mathcal{NP}\subseteq\mathcal{ZPP}$. For a fixed number of columns it remains $\mathcal{NP}$-complete, but there exists a PTAS. The problem can be solved in pseudopolynomial time for a fixed number of rows, and even in polynomial time if all columns furthermore contain entries from the same multiset.