Balanced assignments of periodic tasks

TL;DR

The paper studies balancing periodic task assignments among indistinguishable workers, proving that a balanced feasible assignment exists iff a feasible assignment with every task performed by each worker at least once exists, and that such balanced assignments can be chosen periodic with period $q$. It introduces a pebble-on-arc-colored Eulerian digraph model as a core tool, and shows how to construct periodic balanced solutions in polynomial time for the basic version, with extensions to weekly schedules yielding a larger, explicit period bound. The extended version maintains decidability and balance properties under non-overlapping weekly schedules, though some direct implications from the basic version no longer hold without additional conditions. The framework relies on encoding assignments as directed graphs, transforming instances to unify the analysis, and using merge operations to obtain Hamiltonian cycles that generate periodic balanced schedules. These results have implications for cyclic rosters in transportation and related scheduling domains, providing both constructive algorithms and rigorous balance guarantees.

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.

Figures (10)

• Figure 1: Example of an instance $\mathcal{I}$ with two tasks ($n=2$) and two workers ($q=2$), with a feasible assignment. Up to exchanging workers $1$ and $2$, this assignment is unique and it is not balanced: without loss of generality, the first occurrence of the hatched blue task is assigned to worker $1$, and this determines completely the assignment of all other occurrences of tasks $1$ and $2$.
• Figure 2: Example of an instance $\mathcal{I}$ with three tasks ($n=3$) and three workers ($q=3$), with two feasible assignments.
• Figure 3: An instance $\mathcal{I}$ with four tasks ($n=4$), a set of non-overlapping schedules $\mathcal{S}$, and two workers ($q=2$) showing that the implication $\ref{['cond1']}\Rightarrow \ref{['cond2']}$ given in Theorem \ref{['thm:cns']} does not hold for the extended version.
• Figure 4: An illustration of the transformation of an instance $\mathcal{I}$ into a new instance $\mathcal{I}'$ verifying $|U(\mathcal{I}')|=q$.
• Figure 5: An illustration of a merge operation on a feasible assignment.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof