Continuous-Time Conflict-Based Search (CCBS)

• Continuous-Time Conflict-Based Search (CCBS) is a method for multi-agent pathfinding that defines conflicts via unsafe time intervals and continuous action start times.
• It integrates a high-level constraint tree with a low-level safe interval planner to resolve durative, geometric collisions in continuous environments.
• Recent refinements of CCBS enhance theoretical guarantees and performance, extending its application to real-robot and complex, continuous-time motion planning scenarios.

Continuous-time Conflict-Based Search (CCBS) is a continuous-time adaptation of Conflict-Based Search for multi-agent path finding that replaces discrete vertex- and edge-time conflicts with conflicts between actions executed at specific continuous start times, and replaces point constraints with unsafe time intervals over actions. It was introduced to remove standard assumptions of discretized time, unit-duration actions, and point occupancy, while retaining the two-level CBS decomposition into a high-level constraint tree and a low-level single-agent planner based on safe intervals (Andreychuk et al., 2019). In later literature, CCBS has been described as a leading approach for exact continuous-time MAPF, but its soundness, completeness, and termination guarantees have been re-examined, refined, and in some formulations repaired (Wu et al., 19 Mar 2026, Combrink et al., 22 Aug 2025).

1. Origins and formal setting

The original CCBS formulation studies cooperative pathfinding for non-point, translating, non-rotating agents in 2D workspaces. For exposition, all agents have the same shape and size, move with the same constant speed, and are constrained to the same roadmap graph $G=(V,E)$. Each vertex $v \in V$ is a location an agent may occupy and wait at, and each edge $e \in E$ corresponds to a straight-line trajectory between two locations. Move duration is determined by geometric edge length under constant translational speed, and wait actions may have arbitrary positive real duration (Andreychuk et al., 2019).

A later graph-based formulation that is explicitly used to analyze CCBS in the asynchronous-action setting models the environment as $G=(V,E)$, with each agent $i$ having start $v_s^i$, goal $v_g^i$, and edge traversal times that may depend on both agent and edge. A state is a space-time pair $s^i=(v,t)$, and an action is a transition

$A^i = \big((v_1^i,t_1^i),(v_2^i,t_2^i)\big)$

with duration $\tau(A^i)=t_2^i-t_1^i$. For a move action, $v \in V$0 is adjacent to $v \in V$1 and $v \in V$2 is given by the input; for a wait action, $v \in V$3 and $v \in V$4. This continuous-wait model is central to later critiques of CCBS because it introduces an uncountable family of wait actions (Wu et al., 19 Mar 2026).

CCBS has been used under more than one collision model. In the original roadmap-based presentation, conflicts arise from geometric overlap of moving agents with nonzero volume in continuous time. In the MAPF-AA analysis, the paper assumes Duration Occupancy, under which if agent $v \in V$5 executes

$v \in V$6

then during the whole action it occupies both endpoints; a Duration Conflict occurs whenever two agents occupy the same vertex over a nonempty time interval. This makes CCBS relevant both to geometric continuous-time MAPF and to graph-based asynchronous-action models with durative occupancy (Andreychuk et al., 2019, Wu et al., 19 Mar 2026).

The cost model in the original CCBS paper is sum-of-costs. If $v \in V$7 is the sum of durations of the actions in agent $v \in V$8’s plan, the objective is

$v \in V$9

The MAPF-AA paper likewise develops its replacement method for SoC-optimal planning, with path cost defined as the time an agent reaches its goal and can remain there permanently without conflict (Andreychuk et al., 2019, Wu et al., 19 Mar 2026).

2. Core search mechanics

CCBS preserves the classical CBS architecture. At the high level it maintains a constraint tree whose nodes store a set of constraints and a current joint plan; at the low level it replans one agent at a time subject to those constraints. The root node has no constraints and is formed from individually optimal paths. High-level search is best-first by solution cost, and when a node’s joint plan is conflict-free, that node is returned as a solution (Andreychuk et al., 2019).

The fundamental representational shift is the definition of conflict. In standard CBS, a conflict is typically a vertex-time or edge-time-step event. In CCBS, a conflict is represented as

$e \in E$0

meaning that if agent $e \in E$1 executes action $e \in E$2 at time $e \in E$3 and agent $e \in E$4 executes action $e \in E$5 at time $e \in E$6, then the two agents collide. Because actions are durative and geometry-aware, this representation covers move-move conflicts, move-wait conflicts, and, more generally, continuous-time collisions that are not reducible to synchronized grid occupancies (Andreychuk et al., 2019).

The corresponding constraint is an unsafe-interval constraint on an action. For a conflict $e \in E$7, the unsafe interval of timed action $e \in E$8, denoted $e \in E$9, is the maximal interval starting at $G=(V,E)$0 such that performing $G=(V,E)$1 during that interval creates a conflict with $G=(V,E)$2 performed at time $G=(V,E)$3. The high-level then branches by adding one of the two constraints

$G=(V,E)$4

This is the continuous-time analogue of CBS branching on a pair of mutually exclusive resolutions (Andreychuk et al., 2019, Andreychuk et al., 2021).

The low level in vanilla CCBS is based on Safe Interval Path Planning. A SIPP state is a pair $G=(V,E)$5, where $G=(V,E)$6 is a vertex and $G=(V,E)$7 is a maximal safe interval during which the agent can remain at or arrive at $G=(V,E)$8 without collision. CCBS adapts SIPP so that wait constraints split safe intervals at a vertex, while move constraints forbid departure during unsafe start-time intervals; if a move from $G=(V,E)$9 to $i$0 is forbidden for starts in $i$1, the low-level planner can replace an immediate move by waiting at $i$2 until $i$3 and then moving to $i$4 (Andreychuk et al., 2019, Andreychuk et al., 2021).

Later work on CCBS improvements makes the low-level consequences more explicit. Under disjoint splitting, positive constraints become continuous-time action landmarks of the form “perform action $i$5 sometime in $i$6,” which required a generalized SIPP capable of handling multiple start states, multiple goal states, and sequential landmark satisfaction. That extension was introduced because in continuous time, enforcing the earliest realization of a landmark can be suboptimal or incomplete (Andreychuk et al., 2021).

3. Theoretical guarantees and their revision

The 2019 CCBS paper states that the algorithm “does not rely” on discretized time and uniform action durations, “is complete,” and “provides provably optimal solutions” (Andreychuk et al., 2019). Subsequent work, however, argued that this theoretical picture is incomplete in the presence of arbitrary real-valued waits.

The core critique is that when wait duration is any positive real number, the low-level action space contains an uncountable infinity of waits. In “CBS with Continuous-Time Revisit,” the argument is that if a move-wait conflict is resolved by forbidding a specific wait action in a specific start-time interval, then nearby waits of slightly different duration remain legal and may still conflict. The paper formalizes this by showing that forbidding one wait motion $i$7 can still permit infinitely many variants $i$8 that preserve the conflict, which in turn can force infinitely many expansions before the optimal solution is reached (Li et al., 13 Jan 2025).

A closely related analysis in the MAPF-AA setting states that naively applying CCBS to asynchronous actions leads to incompleteness because waiting has planner-chosen real-valued duration. In that formulation, CCBS branches on constraints tied to a specific wait action and a time interval, but this does not eliminate nearby waits such as $i$9, $v_s^i$0, and $v_s^i$1 when the forbidden action is $v_s^i$2. The paper therefore treats CCBS as practically useful but incomplete in that setting, and further argues that the open-source implementation strengthens wait-action constraints in a way that is not semantically equivalent to the original branch, so it can also lose completeness and optimality (Wu et al., 19 Mar 2026).

A 2025 paper sharpens this critique by separating two issues: the original theoretical variant can suffer from non-termination, while the widely used implementation can return sub-optimal solutions. It introduces an analytical framework in which branching rules are analyzed as pairs of constraint sets, and proves that the public implementation’s move-wait branch is unsound. It then proposes a new move-wait branching rule based on an intersection interval and a parameter

$v_s^i$3

with a move-side branch forbidding starts in $v_s^i$4 and a wait-side branch forbidding both occupancy of the wait vertex and outgoing moves from that vertex during $v_s^i$5. Under that rule, the resulting variant is proved sound and solution complete, and the paper states that it matches the guarantees of discrete-time CBS for the first time in the continuous domain (Combrink et al., 22 Aug 2025).

A different line of theory addresses termination on unsolvable instances. “On the Completeness of Conflict-Based Search: Temporally-Relative Duplicate Pruning” introduces Temporally-Relative Duplicate Pruning for classic and continuous-time MAPF domains. For rational-time continuous MAPF, it defines temporally-relative duplicates via the normalized joint-state map

$v_s^i$6

and proves finiteness of the search space after pruning. This gives termination and completeness for classic MAPF and for the rational-time subset MAPF$v_s^i$7, which is directly relevant to CCBS-style methods on rational-duration domains (Walker et al., 2024).

4. Performance-oriented refinements

Once CCBS had established itself as a practical continuous-time solver, a major research direction became adapting successful CBS enhancements to the continuous-time setting. “Improving Continuous-time Conflict Based Search” introduced three such families of improvements: prioritizing conflicts, disjoint splitting, and admissible high-level heuristics. Because continuous time changes the meaning of conflicts and positive constraints, these adaptations required a generalized SIPP and a refined notion of conflict importance based on cost impact rather than only cardinality (Andreychuk et al., 2021).

In that work, the cost impact of a conflict $v_s^i$8 is defined as

$v_s^i$9

where $v_g^i$0 and $v_g^i$1 are the increases in node cost incurred by resolving the conflict on each side. Two admissible high-level heuristics are then built from these values: an LP-based lower bound with constraints $v_g^i$2, and a greedy disjoint-conflict heuristic that sums cost impacts over pairwise agent-disjoint conflicts. On the reported benchmarks, the enhanced methods increased solved instances by $v_g^i$3, from $v_g^i$4 to $v_g^i$5, and in some settings solved problems with almost twice as many agents as vanilla CCBS while improving runtime by up to two orders of magnitude (Andreychuk et al., 2021).

A second major line of refinement targets symmetry. “Clique Analysis and Bypassing in Continuous-Time Conflict-Based Search” adapts bypassing to CCBS for cost symmetries and introduces biclique- and clique-based reasoning for spatial conflict symmetries. The paper formulates disjoint bicliques and disjoint $v_g^i$6-partite cliques, proves that they preserve completeness and optimality, and reports statistically significant performance improvements over the previous state of the art, solving problems for up to $v_g^i$7 or $v_g^i$8 more agents in the same amount of time on dense graphs (Walker et al., 2023).

Low-level engineering has also become an explicit topic in CCBS-style solvers for nonholonomic agents. “CAR-CHASE” studies a car-like continuous-time CBS variant, CL-CBS, and argues that heuristic values should be treated as context-dependent because CBS constraints alter the effective search space. It introduces conflict-aware heuristic caching keyed by state and a compact conflict fingerprint, together with an adaptive hybrid heuristic that switches between approximate and exact computation. On 480 instances with 10 to 30 agents, the reported geometric mean speedup over the baseline CL-CBS implementation is $v_g^i$9, the success rate rises from $s^i=(v,t)$0 to $s^i=(v,t)$1, total runtime is reduced by $s^i=(v,t)$2, and 33 additional instances are solved within the timeout (To et al., 13 Dec 2025).

5. Variants and application-driven extensions

CCBS has also served as a substrate for variants that broaden the motion model while keeping the high-level conflict-based structure. “Optimal and Bounded Suboptimal Any-Angle Multi-agent Pathfinding” combines CCBS with an any-angle low-level planner, TO-AA-SIPP, to obtain the first optimal any-angle MAPF algorithm. The resulting AA-CCBS preserves CCBS-style unsafe-interval branching but operates over arbitrary line-of-sight segments rather than a fixed local neighborhood. Because any-angle motion greatly enlarges the branching factor, the paper adapts Disjoint Splitting and Multi-Constraints; the resulting enhancements enable solving over $s^i=(v,t)$3 more problems than the vanilla combination of CCBS and TO-AA-SIPP (Yakovlev et al., 2024).

Execution-oriented work has shown that CCBS can be adapted to real robots with modest changes. “Plan Execution for Multi-Agent Path Finding with Indoor Quadcopters” extends an existing CCBS implementation to 3D grids and introduces a cylindrical protection zone around each quadcopter. Collision detection between cylinders is decomposed into an unsafe interval in the $s^i=(v,t)$4-plane for moving circles and an unsafe interval in $s^i=(v,t)$5 for line segments, and the full unsafe interval is their intersection. In a $s^i=(v,t)$6 m flying area with 8 Loco anchors and Crazyflie 2.1 quadcopters, the paper reports successful execution of CCBS-generated plans and tracking errors such as average error $s^i=(v,t)$7 m for the high-level commander mode and maximum error $s^i=(v,t)$8 m for the VLL mode (Kulhan et al., 2022).

Other work develops adjacent continuous-time CBS variants that are useful for understanding the scope of canonical CCBS without being direct implementations of it. A topometric-map method runs CBS over structural-semantic regions with time intervals attached to regions rather than discrete cells, improving corridor-symmetry handling and real-robot deployment but not using canonical CCBS unsafe-interval action reasoning (Fredriksson et al., 29 Jan 2025). CE-CBS moves farther toward continuous geometric motion by combining the CBS high level with RRT* and B-spline smoothing for agents moving along smooth curves in continuous environments (Janovská et al., 2024). These papers suggest that CCBS is part of a broader family of continuous-time CBS-style planners rather than a single fixed implementation.

6. Relation to adjacent continuous-time CBS frameworks

Several papers position themselves explicitly relative to CCBS, either as alternatives for restricted models or as generalizations into richer motion-planning domains. SMT-CBS$s^i=(v,t)$9 reformulates continuous-time CBS as a lazy SAT/SMT process for MAPF$A^i = \big((v_1^i,t_1^i),(v_2^i,t_2^i)\big)$0, retaining continuous conflict validation and unsafe-interval reasoning but replacing the explicit high-level constraint tree with incremental conflict clauses. In that view, the key CCBS idea is not the tree itself but the disjunctive refinement of collision-causing action pairs (Surynek, 2019, Surynek, 2020).

K-CBS extends the CBS architecture directly into kinodynamic multi-robot motion planning without roadmap discretization. Instead of graph actions and unsafe intervals, its conflicts are collision intervals

$A^i = \big((v_1^i,t_1^i),(v_2^i,t_2^i)\big)$1

between continuous trajectories, and its constraints are moving-obstacle constraints induced by another robot’s body along that interval. The method is probabilistically complete rather than optimal, which marks a different design point from canonical CCBS (Kottinger et al., 2022).

A distinct response to CCBS’s continuous-time complexity is to restrict the problem class. CBS-NIC addresses a finite-state model with positive integer edge costs and interval occupancy in discrete time, positioning itself as a CBS-inspired alternative to CCBS for a deliberately restricted problem class. Its motivation is precisely to preserve some duration realism while avoiding what the paper characterizes as CCBS’s unbounded continuous-time state space (Fan et al., 7 Apr 2026).

More generally, CCBS has also been abstracted as an interface pattern rather than a specific algorithm. “Conflict-Based Search as a Protocol” treats CBS as a protocol over a low-level plan() API that returns a path, its occupied space-time volume, and a cost. That paper is explicitly broader than CCBS and does not claim exact continuous-time guarantees, but it illustrates how the CCBS idea of high-level conflict resolution can coordinate heterogeneous planners such as A*, RRT, direct collocation, diffusion models, and reinforcement learning while also making clear that practical space-time constraints are often not mutually disjunctive (Veerapaneni et al., 1 Oct 2025).

Taken together, these works situate CCBS as the canonical continuous-time CBS baseline for roadmap-based MAPF with durative actions, unsafe intervals, and exact high-level branching, while also showing both why it became influential and why its theoretical and algorithmic details have remained active research topics.