Balanced assignments of periodic tasks

TL;DR

This paper investigates the problem of assigning periodic tasks to workers in a balanced manner, ensuring equal task frequency per worker, and provides polynomial-time conditions for existence and extensions with additional constraints.

Contribution

It establishes a necessary and sufficient polynomial-time condition for balanced task assignment and extends results to include additional constraints like work hour limits.

Findings

A polynomial-time verifiable condition for basic balanced assignment.

Existence of periodic balanced assignments under extended constraints.

Tighter bounds on periods for basic balanced assignments.

Abstract

This work addresses the problem of assigning periodic tasks to workers in a balanced way, i.e., so that each worker performs every task with the same frequency over the long term. The input consists of a list of tasks to be repeated weekly at fixed times and a number of indistinguishable workers. In the basic version, the sole constraint is that no worker performs two tasks simultaneously. In the extended version, additional constraints can be introduced, such as limits on the total number of working hours per week. Regarding the basic version, a necessary and sufficient condition for the existence of a balanced assignment is established. This condition can be verified in polynomial time. For the extended version, it is demonstrated that whenever a balanced assignment exists, a periodic balanced assignment exists as well, with a tighter bound on the period for the basic version.