Unmanned Aerial Vehicle Swarm Task Allocation via Temporal Coupling Analysis

Designing efficient task allocation algorithms is paramount for scaling Unmanned Aerial Vehicle (UAV) swarm applications to complex, real-world scenarios. As swarm sizes grow and mission environments become more intricate, traditional methods struggle with the inherent computational complexity and, critically, the impact of temporal task coupling constraints. These constraints, where certain tasks must strictly follow others (e.g., a data retrieval task cannot commence until its associated search task is complete), introduce cross-queue dependencies. Changes in the execution time of a predecessor task (like a search) can cascade, affecting the start times of dependent successor tasks (like data retrieval) potentially assigned to different Unmanned Aerial Vehicles. This leads to frequent allocation failures (where calculated task benefits become invalid) and conflicts, severely degrading solution quality and convergence speed in distributed allocation frameworks. To address these fundamental challenges plaguing Unmanned Aerial Vehicle swarm coordination, this work introduces the Temporal Coupling Analysis-based Task Allocation (TCATA) method.

1. Problem Modeling: Sequential Search and Rescue

Consider a representative sequential task scenario: a post-disaster survivor search and data collection mission. An Unmanned Aerial Vehicle swarm U={u1,u2,…,un}U={u1,u2,…,un} is deployed. Survivors S={s1,s2,…,sm/2}S={s1,s2,…,sm/2} (where mm is even) are randomly located. Each survivor sksk necessitates two sequential tasks:

Search Task (tksearchtksearch): Locate the survivor (performed by a search Unmanned Aerial Vehicle).
Data Task (tkdatatkdata): Collect data (performed by a data Unmanned Aerial Vehicle), which cannot start until tksearchtksearch is completed.
The complete task set is T={t1search,t1data,t2search,t2data,…,tm/2search,tm/2data}={t1,t2,…,tm}T={t1search,t1data,t2search,t2data,…,tm/2search,tm/2data}={t1,t2,…,tm}. The Unmanned Aerial Vehicle swarm is heterogeneous:

Search UAVs: Can only execute search tasks (TTj=1TTj=1).
Data UAVs: Can only execute data tasks (TTj=2TTj=2).

Unmanned Aerial Vehicle Attributes: Each UAV uiui is defined by a tuple:(ui,URi,UTi,UVi,UAi,UDi,UHi,Li)(ui,URi,UTi,UVi,UAi,UDi,UHi,Li)

URi=(xiu,yiu)URi=(xiu,yiu): Initial position.
UTi∈{1,2}UTi∈{1,2}: Type (1: Search, 2: Data).
UViUVi: Maximum cruise speed.
UAi=[ti1,ti2,…,ti∣UAi∣]UAi=[ti1,ti2,…,ti∣UAi∣]: Task execution queue (ordered by start time).
UDi=[TEi1,TEi2,…,TEi∣UAi∣]UDi=[TEi1,TEi2,…,TEi∣UAi∣]: Corresponding task start time list.
UHi⊆UUHi⊆U: Set of neighboring UAVs for communication.
LiLi: Maximum number of tasks uiui can execute.

Task Attributes: Each task tjtj is defined by a tuple:(tj,TSj,TRj,TTj,TDj,TEj,TUj,rj)(tj,TSj,TRj,TTj,TDj,TEj,TUj,rj)

TSj∈STSj∈S: Associated survivor.
TRj=(xjt,yjt)TRj=(xjt,yjt): Task location.
TTj∈{1,2}TTj∈{1,2}: Type (1: Search, 2: Data).
TDjTDj: Task execution duration.
TEjTEj: Task start execution time (to be determined).
TUj∈UTUj∈U: Task executor (to be determined).
rjrj: Earliest possible start time (release time).

Temporal Constraints & Precedence:

The earliest start time rjrj for a data task (TTj=2TTj=2) is the finish time of its associated search task (tjsearchtjsearch) for the same survivor. Search tasks have no explicit release time constraint (rj=0rj=0).
Predecessor (tjpredtjpred) and Successor (tjsucctjsucc) are formally defined:tjpred={∅if TTj=1tkif TTj=2,TSj=TSk,tk≠tj∈T,TTk=1tjpred={∅tkif TTj=1if TTj=2,TSj=TSk,tk=tj∈T,TTk=1tjsucc={∅if TTj=2tkif TTj=1,TSj=TSk,tk≠tj∈T,TTk=2tjsucc={∅tkif TTj=2if TTj=1,TSj=TSk,tk=tj∈T,TTk=2The core constraint is: TEj+TDj≤TEkTEj+TDj≤TEk for all tk=tjsucctk=tjsucc. If a data Unmanned Aerial Vehicle arrives before its associated search task finishes, it must wait.

Objective Function: Given survivors’ critical conditions deteriorating over time, minimizing the sum of all task start times is chosen:min⁡J=∑i=1n∑tj∈UAiTEjminJ=i=1∑ntj∈UAi∑TEj

Subject to:

Temporal Constraint: TEj+TDj≤TEkTEj+TDj≤TEk for all tk=tjsucctk=tjsucc (i.e., ∀TTj=1∀TTj=1).
Resource Heterogeneity Constraint: UTi=TTjUTi=TTj for all ui∈Uui∈U, tj∈UAitj∈UAi (UAV executes only tasks matching its type).
Task Capacity Constraint: ∣UAi∣≤Li∣UAi∣≤Li for all ui∈Uui∈U.

The start time TEjTEj for task tjtj assigned to uiui in position kk of UAiUAi is calculated recursively:TEj=max⁡(TEj−1+TDj−1+dist(TRj−1,TRj)UVi,rj)TEj=max(TEj−1+TDj−1+UVidist(TRj−1,TRj),rj)

where tj−1tj−1 is the preceding task in UAiUAi. For the first task (k=1k=1), TEj−1=0TEj−1=0, TDj−1=0TDj−1=0, and TRj−1=URiTRj−1=URi (UAV’s start position). The dist() function computes the Euclidean distance.

2. Analyzing Temporal Coupling Impact in Market-Based Allocation

Market-based mechanisms (e.g., CBBA) are popular for distributed Unmanned Aerial Vehicle swarm task allocation. UAVs bid on tasks based on perceived utility (often the change in their local cost function if the task is added/removed). Conflict resolution occurs via communication. However, temporal coupling fundamentally disrupts this process.

2.1. Designing the Performance Function under Coupling

Traditional bidding for task tjtj by UAV uiui considers only the impact on its own queue UAiUAi:wi,j⊕l(UAi)=ci(UAi⊕ltj)−ci(UAi)wi,j⊕l(UAi)=ci(UAi⊕ltj)−ci(UAi)wi,j⊖m(UAi)=ci(UAi)−ci(UAi⊖mtj)wi,j⊖m(UAi)=ci(UAi)−ci(UAi⊖mtj)

where ⊕l⊕l denotes insertion at position ll, ⊖m⊖m denotes removal from position mm, and ci(⋅)ci(⋅) is UAV uiui’s local cost (sum of start times of its tasks).

This is insufficient under temporal constraints. Adjusting a search task tjtj (changing its TEjTEj) affects:

Tasks following it in UAiUAi (local effect).
The start time rkrk of its successor data task tk=tjsucctk=tjsucc, potentially assigned to a different Unmanned Aerial Vehicle upup (UApUAp).
Tasks following tktk in UApUAp (remote effect).
Potentially, the start times of tasks dependent on those in (1) or (3) (cascading effect).

Therefore, the true impact (utility wi,jl(A)wi,jl(A)) of adjusting tjtj (inserting into uiui’s queue at ll, removing from its current owner uouo’s queue at mm if assigned) is the net change in the global objective JJ:wi,jl(A)=J(A)−J(A′)=∑uq∈Li,jl[∑tp∈UAqTEp(A)−∑tp∈UAq′TEp(A′)]wi,jl(A)=J(A)−J(A′)=uq∈Li,jl∑tp∈UAq∑TEp(A)−tp∈UAq′∑TEp(A′)

where AA is the current global assignment (all UAqUAq), A′A′ is the assignment after the adjustment, and Li,jlLi,jl is the Affected Queue Set. This set contains the queues of all Unmanned Aerial Vehicles where at least one task’s start time changes due to the adjustment of tjtj. Constructing Li,jlLi,jl involves identifying:

New executor’s queue UAiUAi.
Original executor’s queue UAoUAo (if tjtj was assigned).
Queues containing successors of search tasks located after the insertion point ll in UAiUAi.
Queues containing successors of search tasks located after the removal point mm in UAoUAo (if tjtj was assigned and is a search task).

Algorithm 1 formalizes this construction. Crucially, calculating wi,jl(A)wi,jl(A) requires knowledge of the current global assignment AA and the ability to compute the new start times for all tasks in the affected queues Li,jlLi,jl under the proposed adjustment.

2.2. Performance Value Validity and Failure

In standard market-based algorithms, a UAV must abandon a task (and often subsequent tasks in its bundle) if it loses the bid, as the value of subsequent tasks depended on the now-changed queue. Under temporal coupling, performance value failure is much more frequent and severe due to cross-queue dependencies.

Suppose UAV uaua computes a bid wa,pwa,p for task tptp based on global state AA. Simultaneously, UAV ubub computes a bid wb,qwb,q for task tqtq based on AA. If the adjustment for tptp is executed, moving the system to state A′A′, it likely changes the start times of tasks in Lb,qLb,q (the affected queue set for tqtq’s adjustment). Consequently, the bid wb,qwb,q calculated under AA is likely invalid under A′A′, even if tptp and tqtq are unrelated and assigned to different Unmanned Aerial Vehicles. The bid wb,qwb,q was calculated using now-outdated start times for tasks it depends on indirectly via the temporal chain. Selecting adjustments sequentially based on outdated bids leads to suboptimal or even conflicting allocations.

3. Temporal Coupling Analysis-Based Task Allocation (TCATA)

To overcome the limitations of existing methods for large-scale heterogeneous Unmanned Aerial Vehicle swarms with tight temporal constraints, TCATA employs a distributed three-phase approach per iteration: Local Adjustment Set Construction, Communication & Consensus, and Global Adjustment Set Decision. Its core idea is for UAVs to collaboratively generate a set of potential task re-assignment/addition adjustments, analyze their cross-queue conflicts via temporal coupling, and then optimally select a conflict-free subset to execute simultaneously, minimizing performance failures and communication overhead.

3.1. Phase 1: Local Adjustment Set Construction

Recognizing the prevalence of performance failure, TCATA avoids complex bundle auctions. Each Unmanned Aerial Vehicle uiui constructs a small set PiPi of promising local adjustments. An adjustment pi,jlpi,jl represents inserting task tjtj (either unassigned or currently assigned to another UAV uouo) into uiui’s queue at position ll:pi,jl={wi,jl,tj,ui,uo,l,Li,jl,Ii,jl,Ei,jl}pi,jl={wi,jl,tj,ui,uo,l,Li,jl,Ii,jl,Ei,jl}

wi,jlwi,jl: Performance value (reduction in global JJ, calculated via Eq. above).
tjtj: Task being adjusted.
uiui: New executor.
uouo: Original executor (∅∅ if unassigned).
ll: Insertion position in UAiUAi.
Li,jlLi,jl: Affected Queue Set.
Ii,jlIi,jl: Set of all tasks whose start time changes due to this adjustment.
Ei,jlEi,jl: New start times for tasks in Ii,jlIi,jl.

Construction Process (Algorithm 2):

Generate candidate tasks APiAPi: Tasks tj∉UAitj∈/UAi, matching uiui’s type (UTi=TTjUTi=TTj), and whose predecessor (if a data task) is already assigned (TUjpred≠∅TUjpred=∅). Search tasks are always candidates as they have no predecessors.
For each candidate task tj∈APitj∈APi:
1. For each feasible insertion position ll in UAiUAi (positions 1 to ∣UAi∣+1∣UAi∣+1):
  1. Create temporary global assignment A′A′:
    - Insert tjtj into UAiUAi at position ll.
    - If tjtj was assigned (uo≠∅uo=∅), remove tjtj from UAoUAo.
  2. Construct Li,jlLi,jl (Algorithm 1).
  3. Calculate new start times TEp′TEp′ for all tasks in queues within Li,jlLi,jl using Eq. for TEjTEj recursively.
  4. Calculate performance value wi,jlwi,jl (Eq. above, using TEpTEp from AA and TEp′TEp′ from A′A′).
  5. Record Ii,jlIi,jl (tasks with TEp′≠TEpTEp′=TEp) and Ei,jlEi,jl (their new TEp′TEp′).
  6. Form adjustment pi,jlpi,jl.
2. Select the insertion position ll for tjtj yielding the highest wi,jlwi,jl. Keep this single best adjustment pi,j∗pi,j∗ for tjtj.
Select the top-αα adjustments pi,j∗pi,j∗ with the highest performance values wi,j∗wi,j∗ to form uiui’s Local Adjustment Set Pi={p1,p2,…,pα}Pi={p1,p2,…,pα}.

The parameter αα controls the size of PiPi (typically α=1α=1 or 22), trading off local computation per UAV against the diversity of options for the global decision. Critically, all calculations are based on the current global state AA at the start of the iteration; the local UAiUAi is not updated during this phase.

3.2. Phase 2: Communication & Consensus

To enable conflict-free global decision-making based on a consistent view, all Unmanned Aerial Vehicles must agree on the complete set of proposed adjustments from all swarm members. This forms the Global Adjustment Set CP=⋃i=1nPiCP=⋃i=1nPi.

Consensus Process:

Initialize: Each UAV uiui starts with CPi(0)={Pi,∅,∅,…,∅}CPi(0)={Pi,∅,∅,…,∅} (only knows its own PiPi).
For communication round d=1d=1 to DD (diameter of the communication network UHUH):
1. Each uiui sends its current CPi(d−1)CPi(d−1) to all neighbors uk∈UHiuk∈UHi.
2. Each uiui receives CPk(d−1)CPk(d−1) from all neighbors uk∈UHiuk∈UHi.
3. Each uiui updates CPi(d)CPi(d): For any UAV ujuj, if uiui’s current PjPj is empty (∅∅) but a received CPk(d−1)CPk(d−1) contains a non-empty PjPj (from some neighbor ukuk), adopt that PjPj.
After DD rounds, CPi(D)=CP={P1,P2,…,Pn}CPi(D)=CP={P1,P2,…,Pn} for all ui∈Uui∈U. Consensus is achieved.

To minimize communication bandwidth, only the core identifiers are transmitted: Task ID tjtj, new executor ID uiui, original executor ID uouo, insertion position ll, and performance value wi,jlwi,jl. Upon receiving this data, a UAV reconstructs the full adjustment pi,jlpi,jl using its local knowledge of AA, UVUV, URUR, TRTR, and the reconstruction steps of Algorithm 2 (Steps 9-15).

3.3. Phase 3: Global Adjustment Set Decision via Maximum Weighted Clique

With a consistent CPCP available to all Unmanned Aerial Vehicles, the goal is to select a conflict-free subset S⊆CPS⊆CP of adjustments to execute simultaneously within this iteration, maximizing the sum of their performance values ∑p∈Swp∑p∈Swp. Executing multiple adjustments per iteration drastically reduces total iterations and communication overhead.

Conflict Relation: Two adjustments papa and pbpb conflict (Ra,b=0Ra,b=0) if their execution would affect overlapping sets of tasks. Specifically, if their Affected Queue Sets LaLa and LbLb intersect (La∩Lb≠∅La∩Lb=∅), then executing one changes the start times of tasks the other depends on, invalidating its performance value. If La∩Lb=∅La∩Lb=∅, they are conflict-free (Ra,b=1Ra,b=1).Ra,b={1if La∩Lb=∅(Conflict-Free)0otherwise(Conflict)Ra,b={10if La∩Lb=∅(Conflict-Free)otherwise(Conflict)

Graph Formulation & Solution: The conflict relation defines an undirected graph G=(V,E)G=(V,E):

Vertices VV: Each vertex vpvp corresponds to one adjustment pp in CPCP.
Edges EE: An edge (va,vb)∈E(va,vb)∈E exists iff Ra,b=1Ra,b=1 (adjustments papa and pbpb are conflict-free).
Vertex Weight: The weight of vertex vpvp is the performance value wpwp of adjustment pp.

Selecting a conflict-free set SS with maximal total utility is equivalent to finding a Clique CC in GG (a subset of vertices where every pair is connected by an edge) with the Maximum Weighted Sum (Maximum Weighted Clique – MWC). The selected clique CC represents the optimal conflict-free set of adjustments to execute.

Although MWC is NP-hard in general, the restriction ∣Pi∣=α∣Pi∣=α per UAV makes ∣CP∣=αn∣CP∣=αn. For practical swarm sizes (n≤100n≤100) and small αα, ∣V∣∣V∣ is manageable (e.g., 100-400 vertices). A greedy algorithm provides an efficient, high-quality solution:

Initialize: C=∅C=∅, Ω=VΩ=V (all vertices).
While Ω≠∅Ω=∅:
1. Select vertex vp∈Ωvp∈Ω with the highest weight wpwp.
2. Add vpvp to clique CC.
3. Remove vpvp from ΩΩ.
4. Remove from ΩΩ all vertices not adjacent to vpvp (i.e., conflicting with pp).
Output CC.

Each Unmanned Aerial Vehicle solves this MWC problem locally using the consistent CPCP and conflict graph GG. Since the inputs are identical, all UAVs compute the same clique CC. The adjustments in CC are executed:

Tasks are assigned to their new executors at specified positions.
The start times TEjTEj for all tasks in the combined affected queue set ⋃p∈CIp⋃p∈CIp are updated using the precomputed EpEp values.
Local task queues UAiUAi and start time lists UDiUDi are updated accordingly.

The process repeats (Phases 1-3) until the global task assignment converges (no further beneficial adjustments are found).

4. Simulation Results & Analysis

The performance of TCATA is rigorously evaluated using a sequential search-and-rescue scenario implemented in Python on a distributed simulation platform (Core i5-12600KF, 32GB RAM). Communication latency is modeled at 30ms per round-trip per link; the network is fully connected. TCATA is compared against three benchmarks:

Distributed Genetic Algorithm (DGA): A modified heuristic separating search/data task allocation.
CBBA-TCC: A state-of-the-art market-based algorithm handling temporal constraints.
CNP: A Contract Net Protocol-based approach with deadlock avoidance.

4.1. Algorithm Verification & Parameter Sensitivity

An initial scenario validates TCATA’s feasibility: 50 survivors, 16 Search Unmanned Aerial Vehicles (UT=1UT=1, Li=4Li=4), 24 Data Unmanned Aerial Vehicles (UT=2UT=2, Li=3Li=3), 60 m/s speed, 60s search time, 80s data time, area=18kmx12km. Key results:

Feasibility: All tasks were allocated. All data tasks started after their associated search tasks completed. All UAVs executed tasks matching their type within capacity limits. Paths were collision-free.
Objective: Minimization of sum of start times achieved.

Sensitivity to αα: Table 1 shows how αα (number of adjustments per UAV in Phase 1) impacts performance (averaged over runs). The global objective JJ remains stable.

Table 1: TCATA Performance Sensitivity to Parameter αα

αα	Comp. Time (s)	Iterations	Total Time (s)	Obj. JJ (s)
1	0.151	22	0.811	371.1
2	0.138	14	0.558	371.7
3	0.172	13	0.562	371.2
4	0.212	14	0.632	370.5
5	0.232	11	0.562	372.0
10	0.519	7	0.729	371.5

Computation Time: Increases with αα due to evaluating more adjustments per UAV locally (Phase 1) and solving larger MWC problems (Phase 3).
Iterations: Decreases with αα because more adjustments are considered globally per iteration, allowing more tasks to be allocated per step.
Total Time: Combines computation and communication time (iteration count * comm. latency). α=2α=2 often offers the best balance, minimizing total time. Higher αα reduces iterations but increases computation per iteration.

4.2. Large-Scale Performance Comparison

Scenarios scale up: Survivors (S) / Search UAVs (SU) / Data UAVs (DU): (50/16/24), (100/32/48), (125/40/56), (150/48/64), (175/56/84), (200/64/96). Results are averaged over 30 random instances per scale.

Solution Quality: TCATA consistently produces the best solutions. At 200 survivors, TCATA’s JJ (320.72s) is ~1% better than CBBA-TCC (323.96s), ~8% better than CNP (348.27s), and ~21% better than DGA (407.76s). DGA degrades significantly as problem size increases due to the exploding search space. Market-based methods (TCATA, CBBA-TCC, CNP) leverage spatial/temporal information for better solutions. TCATA’s direct global impact modeling via wi,jlwi,jl and conflict-aware selection provides a slight edge.
Iteration Count: TCATA requires the fewest iterations, scaling best. At 200 survivors, TCATA averages 42.6 iterations, significantly less than CBBA-TCC (107, ~40% of CBBA-TCC) and CNP (452, <10% of CNP). DGA iterations are fixed (300) but its solution quality is poor. CNP’s high iteration count stems from single-task assignments per negotiation round. TCATA and CBBA-TCC assign multiple tasks per iteration, but TCATA’s conflict-aware multi-task selection per MWC is more efficient than CBBA-TCC’s bundle building with potential conflicts requiring removal in later iterations.
Total Execution Time: TCATA is the fastest algorithm, especially at scale. At 200 survivors, TCATA takes 4.38s, significantly faster than CBBA-TCC (21.84s, ~20% of CBBA-TCC), CNP (13.59s, ~32% of CNP), and DGA. This results from:
1. Low Iterations: Minimizes communication rounds and associated latency.
2. Efficient Local Filtering (Phase 1): Focuses global decisions on high-potential adjustments.
3. Conflict-Free Multi-Task Assignment (MWC): Maximizes progress per iteration, avoiding costly rollbacks present in CBBA-TCC.
4. Decoupled MWC Size: The MWC problem size depends only on the number of UAVs (nn) and αα, not the number of tasks (mm), making it scalable.

Table 2: Key Performance Comparison at 200 Survivors (Averages)

Metric	TCATA	CBBA-TCC	CNP	DGA
Objective JJ (s)	320.72	323.96	348.27	407.76
Iterations	42.6	107	452	300
Total Time (s)	4.38	21.84	13.59	>60 (Est.)

5. Conclusion

This work presented the Temporal Coupling Analysis-based Task Allocation (TCATA) algorithm to address the critical challenge of efficient and conflict-free task allocation in large-scale heterogeneous Unmanned Aerial Vehicle swarms under strict temporal constraints. The core contributions are:

Quantified Temporal Coupling: A novel performance function explicitly models the cross-queue impact of task adjustments on the global objective, enabling direct optimization and reducing performance failures prevalent in existing methods.
Distributed Conflict-Aware Allocation: A three-phase approach combining local adjustment generation, consensus-based global set formation, and efficient Maximum Weighted Clique selection for conflict-free multi-task assignment per iteration. This drastically reduces communication rounds and computation time compared to sequential or conflict-prone methods.
Demonstrated Superiority: Extensive simulations in large-scale sequential search-and-rescue scenarios (100+ tasks, 100+ Unmanned Aerial Vehicles) show TCATA achieves:
- Slightly better solution quality than state-of-the-art distributed algorithms (CBBA-TCC, CNP).
- Significantly better solution quality than distributed heuristics (DGA).
- Drastic reductions in iteration count (>60% vs CBBA-TCC, >90% vs CNP).
- Drastic reductions in total execution time (>75% vs CBBA-TCC, >65% vs CNP).

TCATA enables Unmanned Aerial Vehicle swarms to efficiently handle complex, time-sensitive missions by providing fast, high-quality, and scalable distributed task allocation resilient to the challenges of temporal task coupling. Future work involves deeper theoretical analysis of the coupling effects using Markov models or graph theory and exploring Graph Neural Networks for solving the MWC problem to enhance optimality.