Optimization over Young Diagrams

TL;DR

This paper presents a polynomial-time algorithm for optimizing Young diagrams to minimize the sum of evaluations of two functions on conjugate partitions, despite the exponential number of possible diagrams.

Contribution

It introduces a polynomial-time solution to a problem previously thought to be complex due to the exponential number of diagrams.

Findings

Polynomial-time algorithm for Young diagram optimization

Efficient evaluation of conjugate partition functions

Overcomes exponential complexity in diagram selection

Abstract

We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of conjugate partitions. While there are exponentially many diagrams, we show it is polynomial time solvable.