CBSw/P: Conflict-Based Search with Priorities

Updated 30 November 2025

The paper introduces CBSw/P, an algorithm that replaces motion-specific constraints with dynamic priority rules to resolve agent conflicts.
It employs a two-level planning structure where higher-priority agents keep their paths and lower-priority agents replan around them.
CBSw/P scales well in open, high-resolution domains by aggressively pruning unpromising branches, albeit at the cost of completeness and optimality.

Conflict-Based Search with Priorities (CBSw/P) is an algorithmic framework for multi-agent pathfinding (MAPF) that augments classic Conflict-Based Search (CBS) by resolving conflicts via dynamic prioritization—enforcing partial or total orderings among agents—rather than exclusively by motion-specific constraints. CBSw/P systematically searches the space of possible priority orderings, balancing aggressive pruning against theoretical guarantees. The approach is particularly impactful in environments with varying degree of agent coupling, and its aggressive constraint regime facilitates scalability in large and high-resolution domains, albeit typically sacrificing completeness and optimality when compared to conservative constraint-based algorithms.

1. Formalization of Priority Constraints

CBSw/P operates over a state-space defined by a workspace graph $G=(V,E)$ and a set of agents $\mathcal{A} = \{a_1, ..., a_n\}$ . When a collision is detected—via two agents occupying the same vertex or traversing the same edge at the same timestep—CBSw/P replaces the classical motion constraint approach with the introduction of a global priority constraint.

For conflicting agents $a_i$ and $a_j$ , instead of forbidding each from the specific collision, CBSw/P generates two constraint branches:

$\langle a_i \prec a_j \rangle \qquad\text{or}\qquad \langle a_j \prec a_i \rangle,$

where $a_i \prec a_j$ denotes $a_i$ having higher priority than $a_j$ . Under $a_i \prec a_j$ , agent $a_j$ is replanned such that for all $t$ :

$\pi_j[t] \neq \pi_i[t] \quad \wedge \quad \{\pi_j[t] \rightarrow \pi_j[t+1]\} \cap \{\pi_i[t] \rightarrow \pi_i[t+1]\} = \emptyset,$

prohibiting $a_j$ from ever occupying a vertex or edge in $a_i$ ’s assigned path. This mechanism results in “aggressive” pruning and directly encodes dependency by priority, guaranteeing that a lower-priority agent must fully avoid the trajectory of a higher-priority agent (Lee et al., 23 Nov 2025).

2. Algorithmic Structure and Distinguishing Features

CBSw/P retains the two-level architecture of CBS: a high-level Constraint (or Priority) Tree (CT) managing global constraints and a low-level single-agent path planner (typically A*). Key procedural nuances include:

Constraint Branching: On detecting a conflict in CT node $N$ , CBSw/P generates two children: one for each possible priority ordering of the conflicting pair, updating the partial order in each child node.
Low-Level Planning: The agent with higher priority retains its plan, and only the lower-priority agent is replanned under the new constraint, treating higher-priority trajectories as dynamic obstacles.
Branching Factor: While both CBS and CBSw/P branch by conflicts, CBSw/P’s branching is bounded by $O(n^2)$ , related to the number of agent orderings, in contrast to CBS’s potentially much greater $O(2^k)$ expansion in highly coupled cases ( $k$ = total conflicts encountered) (Lee et al., 23 Nov 2025, Ma et al., 2018).

The following table delineates the critical distinctions:

Feature	CBS	CBSw/P
Constraint Type	Motion-specific	Priority ordering
Completeness	Yes	No
Optimality (sum-of-costs)	Yes	No
Space/Branching Complexity	$O(2^k)$ nodes	$O(n^2)$ nodes

3. Theoretical Properties

CBSw/P’s theoretical profile departs from classical CBS, as follows:

Completeness: CBS guarantees completeness whereas CBSw/P is provably incomplete; a globally consistent priority ordering may preclude all feasible solutions in certain instances (Lee et al., 23 Nov 2025, Ma et al., 2018). For “well-formed” MAPF instances—where any agent can wait at start/goal without blocking others—CBSw/P is complete for any total ordering.
Optimality: CBSw/P is not guaranteed to yield optimal (sum-of-costs or makespan) solutions, and can be arbitrarily suboptimal, even on instances where some total order allows the optimal solution. CBS retains optimality under best-first search (Lee et al., 23 Nov 2025, Ma et al., 2018).
Complexity: In high agent density or highly coupled situations ( $k \gg n^2$ ), the dramatic reduction in branching (from $O(2^k)$ to $O(n^2)$ ) means CBSw/P can converge where CBS cannot within practical resource limits.
Empirical Pruning: Many unpromising branches are pruned early in CBSw/P because infeasibility is detected quickly when no conflict-free path exists for a lower-priority agent under dynamic obstacles imposed by higher-priority plans.

4. Empirical Performance and Scalability

Empirical studies highlight CBSw/P’s efficacy in scenarios characterized by high agent count, high spatial resolution, or featureless (“empty”) environments (Lee et al., 23 Nov 2025). Key empirical results include:

On empty 32×32 grids at $R=1$ , CBSw/P solved up to 80 agents (vs. 68 for CBS), with only ≈1.5 % increased sum-of-costs.
CBSw/P demonstrated consistent superiority in “random” map classes and at high agent counts.
In “narrow” or highly coupled topologies, CBSw/P surpasses CBS only once agent count exceeds the low-coupling regime; for small $n$ , CBS is often preferable.
CBSw/P’s aggressive constraints scale better as coupling and resolution increase, whereas CBS's completeness and optimality are retained only at the cost of vastly increased computational burden.

Map Type	CBS Max Agents (R=1)	CBSw/P Max Agents (R=1)
empty-32×32	68	80
maze-32×32	16	20
Berlin_256	88	92
lag303d	24	24

Empirical evaluation also demonstrates that the problem “topology” is decisive in the choice of algorithms: aggressive priority constraints perform best in open, large, or featureless maps; conservative constraints dominate in mixed or bottleneck-dominated spaces (Lee et al., 23 Nov 2025).

5. Relation to Prioritized Planning, Suboptimal CBS, and Variants

CBSw/P generalizes prioritized planning (PP): PP is equivalent to running CBSw/P with a fixed total priority order and splitting only at the root node (Veerapaneni et al., 2022, Ma et al., 2018). Furthermore, the behavior of prioritized planning can be recovered as the special case where the CBS suboptimality parameter $w_{so} \to \infty$ , with conflict-based splitting deferred until prioritized planning fails.

Weighted and suboptimal CBS algorithms such as WF-EECBS and WO-EECBS can interpolate between classical CBS, prioritized planning, and CBSw/P via parameter settings ( $w_h$ , $r$ , $w_{so}$ ), enabling trade-offs between computational work and suboptimality. In particular:

Weighted Focal CBS (W-EECBS): For $w_{so} = \infty$ and $r \to \infty$ , prioritized planning is exactly simulated at the root node; subsequent conflicts are then handled by CBS-based splitting (Veerapaneni et al., 2022).
Empirically, W-EECBS achieves 5–100× reduction in low-level work compared to EECBS, with optimal choice of $r$ dominating speedup performance.

This unifying interpretation enables CBSw/P-based planners to inherit both the scalability of prioritized approaches and the robustness (in principle) of CBS’s conflict-based splitting, serving as a bridge between two historically separate MAPF paradigms.

6. Extensions, Limitations, and Practical Recommendations

The principal limitation of CBSw/P is incompleteness: well-formedness or fortuitous orderings are required for solution finding. For domains where completeness is non-negotiable, conservative motion-based CBS is to be preferred. Suboptimality, especially for highly congested cases, further restricts its use if strict cost bounds are desired (Ma et al., 2018, Lee et al., 23 Nov 2025).

In practical deployment, representation topology, agent density, and computational constraints guide the algorithmic choice:

Large, open, or high-resolution domains with many agents favor CBSw/P for success rate and runtime scalability.
Mixed or bottleneck-rich topologies mandate conservative CBS for solution quality and coverage.
When run-time is constrained or only near-optimal solutions are required, aggressive CBSw/P is recommended.

A plausible implication is that hybrid and dynamically adjustable constraint selection—switching between aggressive priority and conservative motion constraints based on online search feedback and detected coupling—could further optimize performance, but such hybridization is not directly realized in the published methodologies (Lee et al., 23 Nov 2025).

7. Applications Beyond Standard MAPF

CBSw/P and its variants have been adapted to broader planning frameworks, including multi-agent pathfinding among movable obstacles (M-PAMO) (Hu et al., 30 Sep 2025). In this context, prioritized planning (PP-PAMO*) leverages fixed total priority orderings, but its brittleness and suboptimality are amplified by strong agent-obstacle couplings and the dynamics of movable entities.

Empirical results in MAPF among movable obstacles confirm that CBSw/P-derived prioritized approaches are best suited to instances with light coupling and lower box densities, whereas constraint-based (CBS-MOL) techniques are preferable for higher completeness and solution quality requirements, at the expense of computational cost.

In sum, Conflict-Based Search with Priorities offers a systematic methodology for scalable multi-agent pathfinding in suitable environments, interpolating between prioritized and constraint-based paradigms via explicit priority constraints, and serving both as a flexible theoretical bridge and a practical tool for large-scale, high-density MAPF applications (Lee et al., 23 Nov 2025, Veerapaneni et al., 2022, Ma et al., 2018, Hu et al., 30 Sep 2025).