Fair Periodic Assignment

• The Fair Periodic Assignment Problem is a cyclic scheduling challenge that ensures recurring tasks are allocated fairly, with each worker receiving proportional workloads over time.
• It encompasses multiple formulations—including periodic-task, asymptotic frequency, and repeated permutation models—that address distinct fairness criteria using both combinatorial and optimization-based methods.
• Applications in parking systems, Mobility-on-Demand, and reviewer assignments demonstrate how computational trade-offs are managed to achieve balanced, efficient, and practically validated scheduling outcomes.

Searching arXiv for recent and foundational papers on the Fair Periodic Assignment Problem and closely related formulations. arXiv search: "fair periodic assignment problem" The Fair Periodic Assignment Problem denotes a family of repeated allocation and cyclic scheduling problems in which tasks, items, requests, or service opportunities recur over time and must be assigned subject to temporal feasibility and an explicit fairness requirement. In the strict cyclic formulation, a set of periodically repeating tasks is assigned to workers within a repeating schedule, and fairness requires that each worker performs the same work over time (Lieshout et al., 6 Jul 2025). Closely related models impose balanced long-term frequencies for worker–task pairs (Gachet et al., 2024), or require that cumulative allocations already look fair after every prefix of days (Adams et al., 25 Feb 2026). The resulting literature spans combinatorial optimization, fair division, scheduling, transportation systems, and distributed multi-agent control.

1. Formal models and problem scope

A canonical periodic-task model fixes a period $T\in\mathbb{N}$, a set of tasks $\mathcal{I}$, and for each task $i$ a $T$-periodic open interval $(a_i,b_i)$, possibly wrapping around the end of the period. The $r$-th occurrence of task $i$ is $(a_i+rT,b_i+rT)$. A feasible periodic assignment chooses transition arcs $\mathcal{M}\subseteq \mathcal{I}\times\mathcal{I}$ so that each task has exactly one incoming and one outgoing arc; hence every feasible solution decomposes into disjoint directed cycles. Each cycle induces a repeating work pattern, and the classical efficiency objective is to minimize total transition time, equivalently the number of workers required to operate the schedule (Lieshout et al., 6 Jul 2025).

A second formulation represents periodic tasks by intervals $[a_i,b_i)\subset(-1,1)$, with the $\mathcal{I}$0-th occurrence placed at $\mathcal{I}$1. An assignment is a map

$\mathcal{I}$2

where $\mathcal{I}$3 means employee $\mathcal{I}$4 performs the $\mathcal{I}$5-th occurrence of task $\mathcal{I}$6. Feasibility is defined by overlap exclusion: $\mathcal{I}$7 This model makes the long-run frequency of worker–task incidences the central object (Gachet et al., 2024).

A third formulation studies repeated assignments as daily permutations. There are $\mathcal{I}$8 players, $\mathcal{I}$9 indivisible items or chores, and each day one assigns exactly one item to each player using a permutation of $i$0. The cumulative multiset received by player $i$1 after day $i$2 is $i$3. Here the problem is not only to make the final $i$4-day outcome fair, but to ensure fairness after every single day, in a strong ordinal sense (Adams et al., 25 Feb 2026).

Application-driven variants enrich the model with domain-specific state. In parking assignment, drivers or trips are known over a fixed time horizon, parking-lot occupancy evolves over time, and feasibility couples assignment with capacity dynamics (Taheri et al., 2015). In Mobility-on-Demand, requests arrive online, vehicles have time-varying remaining capacity, and assignment decisions are repeated every sampling interval (Liang et al., 2024). These variants preserve the same core structure: repeated assignment under temporal constraints with a fairness criterion defined over trajectories rather than one-shot allocations.

2. Fairness criteria

The literature uses several non-equivalent fairness notions. In the strict periodic scheduling model, fairness is exact and structural: all workers should perform the same work over time. This is formalized by requiring the transition arcs to form a single Hamiltonian cycle, so workers rotate through the same cyclic sequence of tasks, phase-shifted in time (Lieshout et al., 6 Jul 2025). In the asymptotic-frequency model, an assignment is balanced when for every task $i$5 and employee $i$6,

$i$7

This requires every worker to perform every task with the same long-term frequency (Gachet et al., 2024).

Prefix fairness leads to stronger cumulative conditions. In repeated permutation models, top-balance requires

$i$8

while the stronger balance condition requires, for all $i$9,

$T$0

Weak balance relaxes this to

$T$1

These inequalities control the order statistics of each player’s cumulative bundle and imply perpetual ordinal proportionality up to one item (PROP1) under the corresponding assumptions; related sufficient conditions are also given for ordinal PROP2 (Adams et al., 25 Feb 2026).

In application-oriented models, fairness is often expressed through welfare dispersion rather than identical cyclic patterns. A parking formulation defines driver overhead time

$T$2

pairwise envy

$T$3

and the aggregate fairness measure

$T$4

with the intent of making walking times as equal as possible (Taheri et al., 2015). In Mobility-on-Demand, the two fairness criteria are max-min fairness and deviation of utility, where vehicle utility over a horizon segment is

$T$5

and bidding is corrected by local utility history relative to the bidding group (Liang et al., 2024). In reviewer assignment, fairness is envy-based: EF1 for equal demands and WEF1 for non-uniform demands, implemented through picking-sequence mechanisms (Payan et al., 2021).

A common misconception is that “fair periodic assignment” names a single fairness doctrine. The literature instead treats fairness as model-dependent: exact sameness over time, asymptotic balance, prefix-wise ordinal proportionality, envy minimization, max-min utility, deviation reduction, and EF1-type criteria all appear as legitimate formalizations in different settings.

3. Structural results and existence theorems

One of the strongest existence characterizations appears in periodic-task balancing. There exists a balanced feasible assignment if and only if there exists a feasible assignment in which some employee performs every task at least once. Whenever a balanced feasible assignment exists, one can choose it to be periodic, with period at most

$T$6

Moreover, deciding whether a balanced feasible assignment exists can be done in polynomial time, and if $T$7 is fixed then such an assignment can be computed in polynomial time when it exists (Gachet et al., 2024).

In the worker-cycle formulation, the efficiency benchmark is the maximum load

$T$8

The classical periodic assignment problem always has optimum $T$9. The fair periodic assignment problem, which imposes a Hamiltonian cycle, admits a sharp connectivity characterization: an instance admits a fair solution with $(a_i,b_i)$0 workers if and only if the idle interval graph is weakly connected. Consequently, fairness costs either nothing or exactly one extra worker; the fair solution always uses either $(a_i,b_i)$1 or $(a_i,b_i)$2 workers. The normalized statement is

$(a_i,b_i)$3

The same line of work further shows that allowing aperiodic schedules never reduces the price of fairness (Lieshout et al., 6 Jul 2025).

Repeated-permutation fairness yields a different pattern: some fairness notions are universally feasible, while stronger ones are eventually impossible. Top-balanced sequences exist for every positive integer $(a_i,b_i)$4. Balanced sequences exist for all $(a_i,b_i)$5, but there is no balanced sequence of length $(a_i,b_i)$6 for any $(a_i,b_i)$7, and no balanced sequence for any $(a_i,b_i)$8. Weakly balanced sequences still imply perpetual ordinal PROP1, but there is no weakly-balanced sequence of length $(a_i,b_i)$9 for any $r$0. For ordinal PROP2, sufficient balance conditions are known, but universal existence remains open (Adams et al., 25 Feb 2026).

A graph-theoretic periodic scheduling formulation gives local fairness guarantees in terms of graph structure. In the Holiday Gathering Problem, the objective is to find an infinite sequence of independent sets minimizing, for every node $r$1, the maximal gap between two appearances of $r$2. A coloring-based construction yields a period at most

$r$3

for a node of color $r$4, proves this is the best possible asymptotic growth for coloring-based solutions, and also gives a construction with period at most $r$5 for a node of degree $r$6 (Amir et al., 2014).

These results clarify that fairness feasibility depends strongly on the formalization. In some models it is characterized exactly by a connectivity or coverage condition; in others, strong prefix guarantees fail for infinitely many parameter values.

4. Algorithmic methods

Exact algorithms are available in the classical and fair worker-cycle setting. The Shift-Sort-and-Match algorithm computes an optimal periodic assignment in $r$7 time by shifting time so that $r$8, sorting all start and end events, and greedily matching end times to the earliest available start times. For the fair problem, the Patching algorithm starts from an optimal efficient assignment, identifies disjoint cycles, scans transition arcs, and patches overlapping arcs from different cycles; if patching cannot merge all cycles, Nearest Neighbor yields an optimal fair solution using $r$9 workers. The full fair problem is solved exactly in $i$0 time (Lieshout et al., 6 Jul 2025).

The asymptotic-balance formulation uses graph constructions rather than direct time-indexed optimization. A key auxiliary result concerns pebbles moving on an arc-colored Eulerian directed multigraph, where a periodic sequence of colors makes each pebble visit each arc with the same frequency. The schedule construction then proceeds through the digraph $i$1, whose Eulerianity is equivalent to the existence of a feasible assignment with an employee performing every task at least once. Polynomial-time decidability follows from constructing $i$2 via a subroutine reduced to a precoloring extension problem on interval graphs (Gachet et al., 2024).

Optimization-based methods dominate application-driven variants. In fair parking assignment, the base model is a mixed-integer linear program with temporal occupancy recursion, capacity constraints, unique assignment constraints, and driver overhead variables. The direct envy objective is computationally prohibitive: with $i$3, $i$4, and $i$5, CPLEX did not find a feasible solution within 12 hours. The proposed heuristic therefore replaces pairwise envy with a mean-based surrogate, fixes drivers whose current walking times are already close to the mean, and iteratively resolves a reduced MILP until convergence (Taheri et al., 2015).

Distributed repeated-assignment settings use decentralized bidding and dual methods. In Mobility-on-Demand, a modified auction is run with bidding, weight correction, and assignment, using automata-based planning to evaluate each vehicle–request pair through weighted product automata and shortest-path computation. Complementarily, a fairness-aware rebalancing rule relocates idle vehicles toward locations with higher potential utility (Liang et al., 2024). In semi-distributed parking-slot assignment, Lagrangian dual decomposition yields a per-car closed-form choice rule

$i$6

with projected subgradient updates at the central controller and a final feasibility repair step if needed (Alfonsetti et al., 2014).

The algorithmic landscape is therefore heterogeneous: exact combinatorial algorithms for structural periodic models, graph-theoretic constructions for balance, MILP heuristics for time-indexed fairness, and distributed auctions or dual decomposition for online systems.

5. Canonical application domains

Parking systems are an early and prominent application. One formulation treats drivers as known in advance over a daily time horizon and assigns each trip to a parking lot so that occupancy remains feasible over time and the assignment is fair from the driver’s perspective. Lot occupancy evolves according to arrivals and departures, each driver is assigned to exactly one lot, and fairness is driven by walking time from the assigned lot to the destination (Taheri et al., 2015). A related formulation assigns cars to free parking slots at each time slot and minimizes the maximum parking distance across all users, thereby adopting min-max fairness rather than envy-based balance (Alfonsetti et al., 2014).

Mobility-on-Demand introduces a repeated online assignment process. Requests are expressed as co-safe LTL formulas, vehicles evolve on a weighted transition system, and assignments are computed in a distributed manner based only on communication between vehicles. Fairness applies across vehicles rather than customers: the mechanism seeks a fair distribution of accumulated utility while preserving service efficiency (Liang et al., 2024).

Reviewer assignment provides a distinct periodic or sequential interpretation. Papers are treated as agents, reviewers as indivisible goods, and picking-sequence orders determine the repeated order in which papers choose reviewers. The central insight is that fairness comes from the sequence mechanism, while efficiency depends heavily on the chosen order; the resulting algorithms support reviewer capacities, paper demands, conflicts of interest, monotone assignment constraints, and reviewer lower bounds (Payan et al., 2021).

Two further domains illuminate the breadth of the area. The Holiday Gathering Problem asks for an infinite sequence of independent sets in a conflict graph so that each node reappears regularly, linking periodic fairness to graph coloring and prefix-free encodings (Amir et al., 2014). A related periodic scheduling strand studies periodic message assignment on a shared bidirectional link, seeking schedules without contention nor buffering; this line emphasizes schedulability thresholds and polynomial-time algorithms rather than equity across workers or agents (Guiraud et al., 2020). This suggests that fair periodic assignment sits inside a broader class of periodic resource-allocation problems where fairness may or may not be explicit.

6. Empirical performance, trade-offs, and open questions

Empirical studies show that fairness objectives can materially change allocation outcomes. In parking assignment, experiments on New York City taxi data and Cologne TAPAS data use 10 parking lots generated by $i$7-means clustering, 100 sampled drivers, and 500 tests per dataset. Using the proposed minimum-envy heuristic, the reported mean improvement in mean envy is $i$8 over Minimum Sum and $i$9 over No Scheme on NYC data, and $(a_i+rT,b_i+rT)$0 over Minimum Sum and $(a_i+rT,b_i+rT)$1 over No Scheme on Cologne data. Jain’s fairness measure also improves: $(a_i+rT,b_i+rT)$2 and $(a_i+rT,b_i+rT)$3 on NYC, and $(a_i+rT,b_i+rT)$4 and $(a_i+rT,b_i+rT)$5 on Cologne, relative to Minimum Sum and No Scheme respectively; against the Geng/Cassandras-style method, the reported mean relative improvement is $(a_i+rT,b_i+rT)$6 in mean envy and $(a_i+rT,b_i+rT)$7 in Jain’s fairness measure (Taheri et al., 2015).

In Mobility-on-Demand, simulation on a Mid-Manhattan road network with 184 nodes, 20 vehicles, varying request counts from 200 to 400, and auction plus rebalancing every 10 seconds reports that both rebalancing and weight correction increase the minimum utility and decrease utility deviation. The combination of utility-history correction and demand-aware rebalancing gives the strongest fairness gains, while the distributed auction remains competitive with a centralized ILP in average utility and number of unassigned requests (Liang et al., 2024).

Theoretical trade-offs are equally sharp. In the worker-cycle formulation, fairness costs at most one extra worker and aperiodicity never improves that bound (Lieshout et al., 6 Jul 2025). In repeated permutations, the strongest prefix fairness conditions are infeasible for sufficiently large $(a_i+rT,b_i+rT)$8, so perpetual fairness after every day is structurally limited by combinatorial impossibility (Adams et al., 25 Feb 2026). In periodic-task balancing, the clean characterization holds in the common-period and rational-period settings, but if some periods are irrational the theorem can fail (Gachet et al., 2024).

Open questions remain concentrated around the boundary between feasible and infeasible fairness. The repeated-permutation literature leaves open whether, for every $(a_i+rT,b_i+rT)$9, there exists a repeated assignment satisfying the stated sufficient conditions for ordinal PROP2 (Adams et al., 25 Feb 2026). Periodic message assignment leaves the universal load threshold unknown and notes that NP-completeness remains open for the shared-link feasibility problem (Guiraud et al., 2020). Graph-based periodic scheduling conjectures a separation between periodic and non-periodic guarantees under strict local waiting-time bounds (Amir et al., 2014).

Taken together, these results portray the Fair Periodic Assignment Problem not as a single optimization model but as a research program organized around a common question: how to impose fairness on repeating assignments without destroying temporal feasibility or incurring prohibitive efficiency loss. The answer depends on the formal fairness target, the periodic structure of the underlying system, and the degree to which exact cyclic symmetry is required.