# An Optimal Solution for the Muffin Problem

## TL;DR

This paper introduces a recursive algorithm that guarantees optimal solutions for muffin problems, a class of resource division problems related to bottleneck transportation, with implications for matrix division and assignment challenges.

## Contribution

It presents the first recursive algorithm capable of solving any muffin problem optimally, establishing a new method for these resource division problems.

## Key findings

- Algorithm always produces optimal solutions.
- Identifies relationships between problem families.
- Applicable to all muffin problems and related matrix division issues.

## Abstract

The muffin problem asks us to divide $m$ muffins into pieces and assign each of those pieces to one of $s$ students so that the sizes of the pieces assigned to each student total $m/s$, with the objective being to maximize the size of the smallest piece in the solution. Muffin problems are a special type of variant of extended bottleneck transportation problem in which the transportation time is simply the quantity transported between any source and sink and the objective is to maximize the minimum transportation time. Of particular interest are Three Matrix Division and Assignment Problems (3M-DAP), for which all sources have the same supply and sinks are divided into two subsets having the same demand within each subset. Muffin problems are 3M-DAP in which all sinks have the same demand. We present a recursive algorithm for solving any 3M-DAP, and hence any muffin problem, and demonstrate that it always produces an optimal solution. The nature of the recursive algorithm allows us to identify interesting relationships between families of such problems.