Complexity of constructing a balanced feasible assignment when the number of employees is part of the input

Determine the computational complexity of constructing a balanced feasible assignment in the periodic task assignment model where each task i repeats over intervals [a_i + r, b_i + r) with period 1, feasibility forbids overlapping assignments to the same employee, and balance requires that for every task i each employee executes it with long-run frequency 1/q. Specifically, establish whether there exists a polynomial-time algorithm that, given the number of employees q as part of the input and assuming a balanced feasible assignment exists, constructs such an assignment.

Background

The paper proves that deciding existence of a balanced feasible assignment is polynomial-time solvable for the unit-period model and that constructing one is polynomial-time when q is a fixed constant (Theorem 1.3). However, their constructive proof yields a period bounded by q × q!, leading to uncertainty about polynomial-time constructibility when q is part of the input.

Consequently, while the decision problem remains tractable, the status of the construction problem for variable q is explicitly left unresolved, and the authors pose it as a formal question about its complexity.