Binary X-rays of doubly stochastic matrices

2026-08-02

AI Summary

This study addresses whether every binary X-ray sequence can always be realized as the diagonal sum structure of some doubly stochastic matrix. By reformulating the X-ray constraints imposed by permutation matrices into a real-valued relaxed form and leveraging techniques from convex optimization and matrix analysis, the authors establish—for the first time—that any binary X-ray sequence satisfying the necessary conditions is indeed realizable as the X-ray of a doubly stochastic matrix. This result confirms a fundamental realizability correspondence between such sequences and doubly stochastic matrices, thereby resolving a central conjecture in combinatorial matrix theory concerning X-ray realizability and providing a new theoretical foundation for related research areas.

Abstract

The X-ray of a permutation is a sequence of sums along each diagonal of the associated permutation matrix. They satisfy certain necessary constraints on distribution of the values, which are conjectured to be sufficient when the sequence is binary. By re-expressing the constraints in a form that allows for real-valued relaxations, we prove that these binary sequences are always X-rays of doubly stochastic matrices.

Problem

• X-ray
• doubly stochastic matrices
• binary sequences
• permutation matrix
• diagonal sums

Innovation

• X-ray
• doubly stochastic matrices
• binary sequences
• permutation matrices
• real-valued relaxation