Reconstructing permutation matrices from diagonal sums

Abstract

In this paper, we present a result concerning the reconstruction of permutation matrices from their diagonal sums. The problem of reconstructing a sum of k permutation matrices from its diagonal sums is NP-complete. We prove that a simple variant of this problem in which the permutation matrices lie on a cylinder instead of on a plane can be solved in polynomial time. We give an exact, algebraic characterization of the diagonal sums that correspond to a sum of permutation matrices. Then, we derive an O(kn2)-time algorithm for reconstructing the sum of k permutation matrices of order n from their diagonal sums. We obtain these results by means of a generalization of a classical theorem of Hall on the finite abelian groups.