Polynomial-time status of algorithmic results for tasks with different rational periods

Ascertain whether the polynomial-time algorithmic results established for the unit-period model—namely, polynomial-time decidability of existence and, for constant q, polynomial-time construction of a balanced feasible assignment—remain valid when each task i has a possibly different rational period τ_i, with occurrences at [a_i + r τ_i, b_i + r τ_i).

Background

The authors extend the model to allow each task i to have its own period τ_i. They show that when all τ_i are rational, existence of a balanced feasible assignment reduces to the unit-period case, implying the existence theorem (Theorem 1.2) carries over.

However, the reduction may incur a non-polynomial blow-up, so the algorithmic counterpart (Theorem 1.3) is not immediately preserved, and the authors explicitly state that it is unclear whether the polynomial-time results still hold in this generalized setting.