Consensus-Based Bundle Algorithm (CBBA)

• CBBA is a decentralized, auction-based algorithm for dynamic multi-agent task allocation that guarantees conflict-free and convergent task assignments.
• It employs a greedy bundle-building process with local bid vectors to efficiently determine high-utility task allocations and resolve conflicts in real time.
• The CBBA-PR extension introduces controlled partial resets to balance rapid convergence with improved coordination quality, particularly in dynamic, bandwidth-constrained environments.

The Consensus-Based Bundle Algorithm (CBBA) is a decentralized, auction-based algorithm developed for dynamic multi-agent task allocation with strict requirements on speed, robustness, and conflict resolution. CBBA and its extensions, notably those that support partial replanning, are central in domains such as multi-UAV coordination, where new tasks may appear online and global reallocation of all tasks is computationally impractical. The algorithm’s main innovation lies in its bundle-building auction process and consensus-based conflict resolution, which provide guaranteed convergence to conflict-free allocations under constraints of real-time, distributed systems.

1. CBBA Workflow and Key Data Structures

CBBA operates in a decentralized fashion where each agent iteratively (a) constructs a local bundle of tasks according to a greedy scoring (bidding) scheme, and (b) participates in a consensus process to resolve conflicting claims on tasks. Each agent $i$ maintains:

• A bundle $b_{(i)}$ (the ordered list of tasks currently claimed)
• An execution path $p_{(i)}$ reflecting the spatial and temporal order of tasks in its bundle
• A bid vector $y_{ij}$ recording the agent’s bid for each task $j$
• A winner vector $z_{ij}$ indicating, from the agent’s perspective, the identity of the current winner for each task

In every bundle-building iteration, each agent greedily appends the task providing the highest utility increase (or lowest marginal cost) to its bundle, updates corresponding bids, and broadcasts this information to its neighbors. Upon receiving information from peers, each agent reconciles conflicting information, resulting in convergence to a global allocation with the following formal property: every task is assigned to exactly one agent, and no two agents have conflicting assignments.

Enforcing diminishing marginal gains in the greedy ordering is crucial for convergence guarantees; the highest-bid/most-valuable tasks are least likely to be released in subsequent replanning phases.

2. Partial Replanning Extension (CBBA-PR)

In time-sensitive, dynamic missions (such as multi-UAV coordination with online-arriving tasks), it is essential to balance:

• Convergence time: how quickly the system achieves a new conflict-free allocation
• Coordination quality: how well tasks are globally reallocated to optimize a team objective

CBBA-PR introduces a controlled partial bundle reset mechanism to address this trade-off. When a new task $T^*$ arrives, there are several possible response strategies:

Strategy	Bundle Reset Scope	Convergence Time	Coordination Quality
No reset	None	$O(D)$	Low
Partial local reset	Last $n_{i,reset}$ tasks	Fast–moderate	Moderate–high
Partial team reset	Lowest $n_{reset}$ tasks team-wide	Moderate	High (scalable)
Full reset	All tasks	$O(n_t D)$	Highest

where $D$ is the network diameter and $n_t$ is the total number of tasks.

Partial Replanning Algorithmic Formulation:

For every task $j \in \mathcal{J}_{reset}$ to be released by agent $i$, the following updates are performed: $$\begin{aligned} b_i(t) &= b_i(t-1) \ominus j \ p_i(t) &= p_i(t-1) \ominus j \ y_{ij}(t) &= -\infty \ z_{ij}(t) &= -1 \end{aligned}$$ This targeted reset ensures that only the lowest-bid tasks (i.e., those most recently added and usually least critical to the agent’s solution) are released for re-bidding, preserving the high-value prior assignments to the extent possible.

Team-Subset Partial Replanning: CBBA-PR can further restrict participation in the reset to a subteam $\mathbb{I}_{reset}$, reducing communication burden and runtime. The number of tasks to be reset globally is computed as $n_{reset} = t_{response}/(d\times \Delta_{comm})$, where $d$ is the subteam diameter and $\Delta_{comm}$ is a communication delay factor.

3. Trade-Offs: Convergence, Coordination, and Communication

The design of CBBA-PR exposes fundamental trade-offs:

• Resetting fewer tasks ($n_{i,reset}$ small) yields fast convergence, at the cost of suboptimal integration of new tasks, as the solution space is only minimally perturbed.
• Resetting more tasks gives the decentralized auction greater flexibility to reoptimize but increases the number of communication rounds (approaching $O(n_tD)$ for a full reset).
• Resetting must strictly follow the order of lowest bids to preserve the necessary properties for convergence; resetting out-of-order can break the monotonic bid structure, and with it, the algorithm’s convergence proof.

Subteam-based partial replanning strategies further enable scalability by limiting message passing to subgraphs of the agent network. Selecting subteams and computing $n_{reset}$ enable designers to precisely tune the speed-versus-quality tradeoff for mission requirements with strict timing constraints.

4. Implementation Considerations

In decentralized CBBA-PR deployment, agents operate in a network typified by limited bandwidth and incomplete information. Implementation requires careful consideration of:

• Local state reset: Careful management of bundle arrays and winner vectors is needed to minimize loss of valuable assignment information during partial resets.
• Distributed consensus: Agents must adhere to strict broadcast and update protocols to ensure convergence; messages must be timestamped or versioned to resolve inconsistencies.
• Agent coordination: Deciding which subteam of agents participates in each replanning iteration affects convergence speed, runtime, and solution quality, especially in topologies with variable communication delays.

Typical computational requirements are light for individual agents (maintenance of local arrays and message passing), but designers must account for network-induced delays, especially under high agent counts or when operating over multi-hop communication networks.

5. Empirical Evaluation and Performance Benchmarks

Multi-UAV simulations validate CBBA-PR’s performance across a variety of dynamic arrival scenarios. Key empirical findings include:

• No-reset approaches achieve rapid convergence ($O(D)$), but solution quality suffers as new tasks are slotted in isolation, often missing opportunities for improved global assignment.
• Full resets yield the best (greedy-optimal) allocations, but incur high communication overhead and scaling challenges.
• Partial resets (both local and team-wide) provide a principled intermediate regime, empirically achieving substantial improvements in team score compared to no-reset, while incurring only moderate increases in convergence time over the baseline. For some problem instances, partial local resets match or nearly match full reset performance in solution quality.
• Communication efficiency: Team-restricted partial resets empirically reduce the required number of message exchanges and negotiation rounds, benefitting from the linear dependence of convergence on the subteam’s network diameter.

Scalability is thus achieved not only via limited bundle resets, but also by optionally restricting the agents that need to participate in each round, making the approach suitable for large, bandwidth-constrained, or geographically dispersed teams.

6. Relation to Related Algorithms and Theoretical Properties

The CBBA-PR architecture is contingent on the key theoretical property of diminishing marginal gains, as the bundle-building process is inherently greedy. This is shared with other consensus and auction algorithms but is enforced concretely in the ordered bundle, which directly determines the set of tasks eligible for reset. As shown by supporting analyses, breaking the monotonicity (for example, by releasing high-bid tasks before low-bid ones) can destroy algorithm guarantees.

CBBA’s consensus-based mechanisms are distinct from continuous-time consensus optimization or mean-field convergence techniques used in other swarm-based or multi-agent optimization algorithms. However, the decomposition of a large assignment problem into local, conflict-resolving exchanges provides generalizable design principles for distributed optimization in resource-allocation, network, and control problems.

7. Applications and Extensions

The primary application for CBBA and its partial replanning extension is dynamic multi-agent task allocation, particularly in time-constrained, communication-limited environments (e.g., UAV teams, disaster response, and mobile robotics). The adjustable parameters of the partial reset (both in number of tasks and in subset of agents) allow mission planners to tune system behavior to priority axes such as minimized latency, maximized global utility, or minimal communication burden.

CBBA-PR forms a foundation for subsequent extensions, including payload-augmented bundle allocation, reputation-based resilient consensus for adversarial environments, and hybrid schemes incorporating parallel evaluation or learning-driven utility estimation. The CBBA-PR protocol has also served as an architectural reference for distributed system designs beyond robotics, such as federated optimization and dynamic cloud resource allocation, wherever decentralized rapid coordination under online arrival of objectives is essential.