The n-Card Problem, Stochastic Matrices, and the Extreme Principle

The n-card problem is to determine the minimal intervals [u, v] such that for every n×n stochastic matrix A there is an n×n permutation matrix P (depending on A) such that tr(PA) ∈ [u, v]. This problem is closely related to classical mathematical problems from industry and management, including the linear assignment problem and the travelling salesman problem. The minimal intervals for the n-card problem are known only for n 6 4. We introduce a new method of analysis for the n-card problem that makes repeated use of the Extreme Principle. We use this method to answer a question posed by Sands (2011), by showing that [1, 2] is a solution to the n-card problem for all n > 2. We also show that each closed interval of length n n−1 contained in [0, 2) is a solution to the n-card problem for all n > 2.