Determine the maximum permanent at fixed average row and column sum

Determine the maximum permanent of a square (0,1)-matrix when the average number of ones in each row and in each column is a prescribed value k.

Background

Problem 1.3 asks for the extremal permanent when only the average number of ones per row and per column is fixed, rather than requiring every row and column to have the same number of ones. This formulation generalizes the setting of Ryser’s conjecture and is equivalent to maximizing the permanent over classes U(n, τ) with a prescribed total number of ones.

The paper explicitly states that this problem has not been solved so far. Its results address only particular parameter ranges and therefore do not resolve the general fixed-average problem.