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

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. [Maximally recoverable codes for grid-like topologies, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 2092-2108] 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 Kn, n, with labels coming from Fd2 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 × 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