# The independence number of the Birkhoff polytope graph, and applications to maximally recoverable codes

## TL;DR

This paper establishes that the minimal dimension for labeling edges in a bipartite graph to ensure cycle sum conditions is linear in n, improving bounds and applying representation theory to analyze the Birkhoff polytope graph.

## Contribution

It proves that the minimal label dimension is linear in n, refining previous bounds, and introduces a recursive construction alongside representation theory analysis.

## Key findings

- The minimal label dimension d is linear in n.
- A recursive construction outperforms random methods.
- Tight bounds for the independence number of the Birkhoff polytope graph are provided.

## Abstract

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph $K_{n,n}$ with labels coming from $F_2^d$ , that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n x n grid topologies. Prior to the current work, it was known that d is between $(\log n)^2$ and $n\log n$. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.