Iterative Exchange for MUCPP

Updated 6 December 2025

The paper introduces an iterative exchange framework that optimizes multi-UAV path planning by balancing overall route efficiency and individual workload fairness.
It employs LPT-style task assignment, 2-opt heuristics, and a series of local exchanges to iteratively reduce a composite efficiency-fairness objective under safety and feasibility constraints.
Empirical results on realistic 3D terrain datasets show reduced composite costs and makespans, demonstrating improved predictability and robustness compared to traditional methods.

The Iterative Exchange Framework for Multi-UAV Cooperative Path Planning (MUCPP) is a recent algorithmic paradigm designed to explicitly balance mission efficiency and agent fairness in multi-vehicle motion planning. It systematically improves team trajectories by iteratively reassigning and reordering task allocations among UAVs, refining multi-agent solutions under feasibility, safety, and dynamics constraints. Below, the core principles, mathematical underpinnings, and applied results of this framework are presented as established by the primary literature (Li et al., 29 Nov 2025).

1. Problem Definition and Theoretical Motivation

The MUCPP problem, as considered in the Iterative Exchange Framework, seeks to generate collision-free trajectories for a team of $K$ UAVs starting from known initial locations $S = \{s_1, ..., s_K\}$ . A finite set of $N$ tasks $T = \{t_1, ..., t_N\}$ must be serviced, each assigned to exactly one UAV and visited precisely once. Every UAV’s route $\pi_k$ is an ordered sequence starting from $s_k$ and covering its assigned tasks. The framework’s key objectives are:

Efficiency: Minimize the total mission route length $L_{\text{total}} = \sum_{k=1}^K L(\pi_k)$ .
Fairness: Minimize the maximum individual route length (makespan) $L_{\max} = \max_{1 \leq k \leq K} L(\pi_k)$ .

The composite weighted objective is: $\min_{\,\Pi}\;J(\Pi)\;=\;\alpha\sum_{k=1}^K L(\pi_k)\;+\;(1-\alpha)\,\max_{k=1\ldots K}L(\pi_k)$ where $\alpha \in [0,1]$ trades off between efficiency and fairness. Solutions must satisfy:

Each task assigned exactly once,
Collision-free, obstacle- and terrain-aware paths,
Altitude constraints above local terrain $z(x, y) \geq h(x, y) + H_{\text{safety}}$ .

This formalism reflects mission-critical multi-agent inspection and logistics scenarios where both overall system economy and agent workload balancing are required (Li et al., 29 Nov 2025).

2. Algorithmic Structure

The framework employs a sequence of constructive initialization followed by iterative local exchanges and re-optimization steps:

Initialization (LPT-Style Assignment and 2-Opt):
- Each task $t_j$ is assigned to the route $\pi_k$ that yields the smallest increase in $J$ , in order of decreasing criticality using a Longest-Processing-Time (LPT) heuristic.
- Routes are ordered using nearest-neighbor or open-path 2-opt heuristics to minimize individual costs.
Iterative Local Exchanges:
- Main improvement step: for up to $I_{\text{max}}$ $I_{max}$ sweeps (typically hundreds to thousands), repeatedly consider the following operations across routes:
  - Relocate: Move a task from one route to another at the lowest cost insertion position.
  - Swap: Cross-exchange two tasks between routes.
  - Intra-route 2-opt: Reverse subsequences within a route.
  - Cross-exchange: Exchange subchains between routes.
  - Kernighan–Lin (KL) Rebalancing: Migrate boundary tasks between critical routes.
- Each move is accepted only if it strictly decreases $J$ , with tie-breaking for makespan and route length. Efficient caching and focus on “critical” (maximal load) routes restrict candidate set sizes.
Final Path Refinement:
- For each optimized task order, A* search is used in a terrain-aware configuration space $(x, y, z)$ to construct actual collision-free, altitude-constrained trajectories.

This approach maintains feasibility at all stages. During inner iterations, a cached or heuristic estimate of segment costs is typically used to accelerate evaluation; full A* refinement is deferred to the terminal routes (Li et al., 29 Nov 2025).

3. Terrain-Aware Path Planning and Feasibility

The distance oracle $D(p, q)$ is computed by a terrain-aware A* algorithm:

State space: $(x, y, z)$ ordered lattice; at each $(x, y)$ , only heights satisfying $z \geq h(x, y) + H_{\text{safety}}$ are permitted.
Obstacle cells are masked out.
Cost per move is Euclidean; heuristic is straight-line (admissible).
The sequence to cover each route is reconstructed as an admissible 3D path after the exchange operations conclude.

Feasibility constraints automatically enforce terrain clearance and obstacle avoidance upon path refinement. This separation between rapid combinatorial exchange (using distance oracles) and full geometric planning is crucial for computational efficiency (Li et al., 29 Nov 2025).

4. Computational Complexity and Convergence

Move Evaluation: Each local exchange modifies $O(1)$ edges and can be updated in $O(1)$ - $O(\log N)$ time using cached segment lengths.
Candidate Moves: $O(N)$ – $O(N \log N)$ per sweep, focused on critical and adjacent routes in task-space.
Total Iterations: $\leq I_{\text{max}}$ (hundreds–thousands); in practice, convergence to no further $J$ improvement is achieved in several hundred.
Final Stitching: $K$ calls to A*; computational cost is dominated by grid resolution but amortized over route length.

The framework is polynomial per sweep but overall complexity is pseudo-polynomial due to the NP-hardness of the underlying Multi-Depot VRP/task allocation. Empirical convergence is obtained well before $I_{\text{max}}$ (Li et al., 29 Nov 2025).

5. Empirical Results and Performance

Extensive benchmarks were performed on ten realistic 3D terrain datasets, varying both $K$ (5–10) and $N$ (50–200). Metrics evaluated:

$J(\Pi)$ : composite efficiency-fairness objective,
$L_{total}$ : total route length,
$L_{max}$ : makespan.

Algorithm	$J(\Pi)$	$L_{total}$	$L_{max}$
Hungarian-Insertion A*	1443.34±205.95	1781.77±156.54	1104.92±326.14
LPT-Balanced A*	1366.90±116.86	2003.93±194.43	729.87±52.76
Iterative-Exchange A*	1262.70±75.38	1773.25±167.57	752.15±62.15

The Iterative-Exchange method attained the lowest average composite objective and makespan, while maintaining competitive total route lengths compared to specialized baselines.
3D route visualizations demonstrate uniformly distributed and compact UAV task assignments.
Box-plots for $J$ reveal lower variance under the proposed method, indicating improved predictability and robustness in solution quality.

These results confirm that iterative local exchanges—when orchestrated under a balanced composite objective—can simultaneously achieve efficiency and fairness superior to approaches focusing on a single metric (Li et al., 29 Nov 2025).

6. Limitations and Future Research Directions

Current limitations of the framework include:

All tasks and obstacles are static; the method does not yet address online replanning under dynamic changes.
Energy consumption models and explicit UAV communication constraints are not integrated.
The approach is centralized; scalability for very large teams may require distributed or hierarchical variants.

Proposed directions for further development:

Extension to dynamic MUCPP accounting for moving obstacles or time-varying tasks.
Incorporation of flight energy/fuel models and limited communications into the objective and constraints.
Development of parallelized or decentralized iterative-exchange algorithms for scaling to large heterogeneous multi-agent systems (Li et al., 29 Nov 2025).

7. Comparative Context, Generalizations, and Impact

The Iterative Exchange Framework sits within the broader trajectory optimization literature as an advanced form of combinatorial multi-agent task allocation and motion planning. Key differentiators relative to other frameworks and heuristics:

Explicit composite objective incorporating both mission efficiency and fairness.
Systematic local exchanges and cross-exchanges, inspired by VRP, TSP, and classic load-balancing (e.g., KL, LPT) heuristics, refined for high-dimensional multi-agent settings.
Deferred full geometric path construction, leveraging fast proxy distances during combinatorial search.

Unlike decentralized agreement protocols (Zhao et al., 2018) which focus primarily on energy under distributed coordination, or leader-follower decompositions (Wu et al., 2019), the Iterative Exchange Framework generalizes to obstacle-rich, terrain-aware global planning with flexible workload allocation and direct terrain/obstacle integration.

In summary, the Iterative Exchange Framework provides a computationally tractable, empirically validated strategy for jointly optimizing efficiency and fairness in multi-UAV cooperative path planning, with a clear path toward extensions in dynamic, energy-aware, and scalable multi-agent domains (Li et al., 29 Nov 2025).