Monotone Optimal Multipartitions Using Schur Convexity with Respect to Partial Orders

Abstract

In a $( t,n,m )$-multipartitioning problem, t lists of $nm$ ordered numbers are partitioned into n sets, where each set contains m numbers from each list. The goal is to maximize some objective function that depends on the sum of the elements in each set and is called the partition function. The authors use the recently developed theory of majorization and Schur convexity with respect to partially ordered sets to study optimal multipartitions for the above problem. In particular, they apply the results to construct a class of counterexamples to a recent conjecture of Du and Hwang, which asserts that (classic) Schur convex functions can be characterized as the partition functions for $( 1,n,m )$-multipartitioning problems having monotone optimal solutions.