A-CMTS: Anytime Congestion Mitigation Tree Search

• A-CMTS is a scalable solver for the congestion mitigation path planning problem, defined on sparse directed graphs with multiplicative vertex penalties.
• It employs a two-layer search architecture akin to CBS, using forced and forbidden edge commitments to manage congestion and provide bounded-suboptimality guarantees.
• The method integrates with local collision avoidance systems (e.g., ORCA, PIBT), achieving significant runtime and throughput improvements over exact MINLP approaches.

Searching arXiv for the target paper and closely related CMPP work to ground the article. A-CMTS, short for Anytime Congestion Mitigation Tree Search, is a scalable solver for the Congestion Mitigation Path Planning (CMPP) problem introduced for large-scale multi-agent navigation in dense environments (Kato et al., 7 Aug 2025). In CMPP, agents are assigned coarse-level, time-independent routes on a sparse directed graph, while local collision avoidance is handled independently by methods such as ORCA in continuous space and PIBT in discrete space (Kato et al., 7 Aug 2025). A-CMTS addresses the nonconvexity induced by a flow-based multiplicative penalty at each vertex, where congestion grows steeply when frequently traversed paths intersect through multiple incoming edges, and it does so through a two-layer search akin to CBS with an explicit suboptimality parameter $\omega \ge 1$ (Kato et al., 7 Aug 2025).

1. Problem setting and congestion model

The CMPP instance solved by A-CMTS is defined on a sparse directed graph $G=(V,E)$ with bidirectional edges, a set of agents $A=\{1,\ldots,n\}$, and start/goal pairs $S=(s_1,\ldots,s_n)$ and $G=(g_1,\ldots,g_n)$, where $s_i,g_i\in V$ (Kato et al., 7 Aug 2025). A solution $\pi=\{\pi_1,\ldots,\pi_n\}$ assigns each agent $i$ a simple path $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$ satisfying start/goal, adjacency, and simple-path constraints (Kato et al., 7 Aug 2025).

The defining feature of CMPP is that congestion is modeled at vertices rather than as an additive edge cost. For each directed edge $e=(u,v)\in E$, the edge flow is

$G=(V,E)$0

The congestion degree at vertex $G=(V,E)$1 is

$G=(V,E)$2

where $G=(V,E)$3 denotes the set of edges entering $G=(V,E)$4 (Kato et al., 7 Aug 2025). The objective is to minimize

$G=(V,E)$5

This multiplicative construction makes vertices where many agents enter from diverse directions become very expensive (Kato et al., 7 Aug 2025). The illustrative example in the paper is deliberately extreme: if 20 agents all use one incoming edge, then $G=(V,E)$6; if the same 20 agents are distributed equally over four incoming edges, then $G=(V,E)$7 (Kato et al., 7 Aug 2025). The formulation therefore discourages multi-directional merges rather than mere volume on a single approach.

This places A-CMTS within a broader CMPP line that treats congestion as a first-class planning signal rather than a post hoc consequence. Related formulations distribute robots across topologically distinct routes (Wang et al., 2022), bias single-agent search using predictive space-utilization heuristics (Han et al., 2021), or reduce high-level cell congestion through hierarchical flow planning (Pan et al., 2024). A-CMTS is distinguished by embedding the congestion penalty directly into a static routing objective on a sparse graph (Kato et al., 7 Aug 2025).

2. Optimization structure and exact formulation

The full CMPP model is a mixed-integer nonlinear program. Its decision variables are $G=(V,E)$8 for agent-edge usage, with auxiliary variables $G=(V,E)$9 and $A=\{1,\ldots,n\}$0 (Kato et al., 7 Aug 2025). The optimization problem is

$A=\{1,\ldots,n\}$1

subject to

$A=\{1,\ldots,n\}$2

$A=\{1,\ldots,n\}$3

$A=\{1,\ldots,n\}$4

$A=\{1,\ldots,n\}$5

$A=\{1,\ldots,n\}$6

The critical source of difficulty is the constraint

$A=\{1,\ldots,n\}$7

which makes the problem nonconvex (Kato et al., 7 Aug 2025). The paper implements an exact MINLP solver in SCIP, using branch-and-bound over $A=\{1,\ldots,n\}$8 with nonlinear constraints enforced via outer approximation (Kato et al., 7 Aug 2025). Under a 60 s limit, this solver succeeds on small instances but does not scale to larger ones. On a $A=\{1,\ldots,n\}$9 Grid, success is 100% for 10, 20, and 30 agents, with objective values 28.0, 47.7, and 70.4, and times 0.8 s, 2.8 s, and 19.0 s, respectively; on a $S=(s_1,\ldots,s_n)$0 Grid, the solver Timeouts at 100 and 200 agents, and N/A is reported for 300 agents (Kato et al., 7 Aug 2025).

A-CMTS is introduced precisely against this backdrop: it is the scalable alternative to the exact MINLP, retaining a formal approximation guarantee through $S=(s_1,\ldots,s_n)$1 while avoiding direct global solution of the nonconvex program (Kato et al., 7 Aug 2025).

3. Two-layer search architecture

A-CMTS is described as a two-layer search akin to CBS (Kato et al., 7 Aug 2025). Its high-level search maintains nodes $S=(s_1,\ldots,s_n)$2 storing

• $S=(s_1,\ldots,s_n)$3: forced $S=(s_1,\ldots,s_n)$4 constraints,
• $S=(s_1,\ldots,s_n)$5: forbidden constraints,
• $S=(s_1,\ldots,s_n)$6: current set of paths,
• $S=(s_1,\ldots,s_n)$7,
• $S=(s_1,\ldots,s_n)$8: a lower bound satisfying

$S=(s_1,\ldots,s_n)$9

The high-level algorithm initializes the root node with empty $G=(g_1,\ldots,g_n)$0 and $G=(g_1,\ldots,g_n)$1, computes the root solution using Prioritized Planning (PP) via low-level, and inserts the node into a priority queue Open ordered by cost (Kato et al., 7 Aug 2025). The incumbent solution is tracked through $G=(g_1,\ldots,g_n)$2 and $G=(g_1,\ldots,g_n)$3.

Node selection and branching are driven by the most congested unresolved vertex. Specifically, A-CMTS selects

$G=(g_1,\ldots,g_n)$4

where

$G=(g_1,\ldots,g_n)$5

It then selects an agent $G=(g_1,\ldots,g_n)$6 and an incoming edge $G=(g_1,\ldots,g_n)$7 satisfying

$G=(g_1,\ldots,g_n)$8

and expands node $G=(g_1,\ldots,g_n)$9 into two children (Kato et al., 7 Aug 2025). Child $s_i,g_i\in V$0 adds $s_i,g_i\in V$1 to $s_i,g_i\in V$2 and leaves the paths unchanged; child $s_i,g_i\in V$3 adds $s_i,g_i\in V$4 to $s_i,g_i\in V$5, then replans $s_i,g_i\in V$6 via the low-level and also replans $s_i,g_i\in V$7 for other agents $s_i,g_i\in V$8 that visit $s_i,g_i\in V$9 (Kato et al., 7 Aug 2025). Only the affected parts of the solution are recomputed.

The anytime property comes from iterative refinement of the incumbent solution under time limits. The suboptimality parameter $\pi=\{\pi_1,\ldots,\pi_n\}$0 controls pruning: if

$\pi=\{\pi_1,\ldots,\pi_n\}$1

node $\pi=\{\pi_1,\ldots,\pi_n\}$2 is pruned (Kato et al., 7 Aug 2025). This produces the characteristic trade-off noted in the paper: higher $\pi=\{\pi_1,\ldots,\pi_n\}$3 $\pi=\{\pi_1,\ldots,\pi_n\}$4 more pruning $\pi=\{\pi_1,\ldots,\pi_n\}$5 faster but looser bound (Kato et al., 7 Aug 2025).

A plausible implication is that A-CMTS preserves the conflict-directed structure of CBS while replacing collision conflicts with congestion-inducing edge commitments. This suggests that the search tree is organized around explanations for high $\pi=\{\pi_1,\ldots,\pi_n\}$6 values rather than spacetime collisions.

4. Low-level search and approximation guarantee

The low-level problem for a single agent $\pi=\{\pi_1,\ldots,\pi_n\}$7 under per-agent constraints $\pi=\{\pi_1,\ldots,\pi_n\}$8 and $\pi=\{\pi_1,\ldots,\pi_n\}$9 is to find $i$0 minimizing incremental congestion:

$i$1

subject to $i$2, $i$3, $i$4, and all $i$5 must appear in $i$6 (Kato et al., 7 Aug 2025). The implementation uses an approximation in which forced edges are visited in order of Euclidean distance, and Dijkstra on $i$7 provides the route between them (Kato et al., 7 Aug 2025).

The solver’s formal guarantee is stated explicitly: A-CMTS returns solution cost $i$8 (Kato et al., 7 Aug 2025). The proof sketch given is that any branch excluding the optimal path is pruned only when

$i$9

Setting $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$0 ensures optimality (Kato et al., 7 Aug 2025). In this sense, A-CMTS is not merely heuristic search; it is an anytime bounded-suboptimal method with a tunable approximation factor.

The paper also places A-CMTS in relation to established paradigms. Compared with MAPF, which is time-dependent, collision-free grid paths minimizing sum-of-lengths, CMPP is time-independent coarse routing minimizing a congestion penalty (Kato et al., 7 Aug 2025). Compared with min-cost flow, where costs are additive on edges, convex, CMPP uses multiplicative vertex costs across incoming edges, nonconvex (Kato et al., 7 Aug 2025). Compared with online collision avoidance such as ORCA/RVO and PIBT, A-CMTS supplies global guidance while leaving fine-scale conflicts to reactive controllers (Kato et al., 7 Aug 2025). This division of labor is conceptually close to hierarchical congestion-aware frameworks in large-scale replanning (Pan et al., 2024), but the objective structure is distinct.

5. Integration with local collision avoidance and operational use

A-CMTS produces coarse-level, time-independent routes on the sparse graph $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$1 (Kato et al., 7 Aug 2025). These routes are not themselves collision-free trajectories in continuous time. Instead, they are designed to be consumed by local control systems.

For continuous-space deployment, the paper describes integration with ORCA. A CMPP path $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$2 is computed, a waypoint queue

$\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$3

is formed, and ORCA steers the agent toward the head of $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$4; when the agent comes within threshold $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$5, that waypoint is popped and the agent proceeds to the next one (Kato et al., 7 Aug 2025).

For discrete deployment, the paper integrates with PIBT in lifelong-MAPF settings. A sparse graph $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$6 is constructed by sampling grid vertices $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$7. Each agent’s next waypoint is the representative grid cell $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$8, or the goal if $\pi_i=(\pi_i[1],\ldots,\pi_i[L_i])$9. PIBT then plans one-step collision-free moves toward that waypoint. Agent positions on the sparse graph are updated via $e=(u,v)\in E$0, and the visited vertex is removed from $e=(u,v)\in E$1, which can be reused as the initial solution for the next CMPP solve (Kato et al., 7 Aug 2025).

This architecture is consistent with other CMPP strategies that separate a global congestion-aware planner from local execution. Communication-free topological routing disperses robots over path classes while relying on local replanning and reactive control (Wang et al., 2022); swarm planning through Virtual Tubes similarly uses centralized path generation and decentralized control inside the tube (Mao et al., 2024). A-CMTS occupies the same systems layer, but with a vertex-penalty objective designed for dense graph-based traffic (Kato et al., 7 Aug 2025).

6. Empirical performance, scope, and limitations

The paper reports that A-CMTS scales empirically to $e=(u,v)\in E$2 up to 2,500 and $e=(u,v)\in E$3 up to 10,000 within seconds (Kato et al., 7 Aug 2025). This is the central practical result: the solver is intended for regimes far beyond what the exact MINLP can handle.

On the $e=(u,v)\in E$4 Grid with $e=(u,v)\in E$5, A-CMTS ($e=(u,v)\in E$6) improves the objective from init cost = 90.7 to final = 74.4, an 18.0% improvement, within 60 s (Kato et al., 7 Aug 2025). On the $e=(u,v)\in E$7 Grid with $e=(u,v)\in E$8, where the MINLP times out at $e=(u,v)\in E$9, A-CMTS achieves 100% success, produces initial solutions in <0.1 s, and yields final solutions ~7% better (Kato et al., 7 Aug 2025). On lak303d with $G=(V,E)$00, where the MINLP fails beyond $G=(V,E)$01, A-CMTS yields ~9% cost reduction over initial (Kato et al., 7 Aug 2025).

The downstream effect of CMPP-guided routing is evaluated in both continuous and discrete settings. In the continuous-space ORCA experiments on an environment of size $G=(V,E)$02 with agent radius $G=(V,E)$03 and horizon $G=(V,E)$04, both vanilla ORCA and CMPP-guided ORCA achieve 100% success by 40 s at $G=(V,E)$05; at $G=(V,E)$06, vanilla ORCA reaches 100% at ~60 s while CMPP-guided ORCA reaches 100% at ~40 s; at $G=(V,E)$07, vanilla ORCA attains 83.9% at 60 s, whereas CMPP-guided ORCA reaches 99.0%, a gain 15.1 pp (Kato et al., 7 Aug 2025). In discrete lifelong-MAPF, the reported throughput gains include +58.1% on warehouse-10-20-10-2-1 with $G=(V,E)$08 and +15.7% on random-64-64-10 with $G=(V,E)$09, while lak303d shows a slight reduction due to unavoidable single-lane bottlenecks (Kato et al., 7 Aug 2025).

Runtime allocation also matters. Increasing the A-CMTS runtime limit $G=(V,E)$10 from $G=(V,E)$11 to $G=(V,E)$12 improves PIBT throughput by up to 10% (Kato et al., 7 Aug 2025). This suggests that the anytime refinement mechanism is operationally useful even under strict control-loop budgets.

The limitations stated in the paper are structural rather than incidental. A-CMTS assumes static start/goal sets per CMPP solve; its time-independent routes may not adapt immediately to dynamic changes; and it relies on local collision avoidance to resolve fine-scale conflicts—no deadlock guarantee if local method fails (Kato et al., 7 Aug 2025). The cited future directions are to integrate dynamic task allocation, product-placement optimization, robustness to unpredictable agent behaviors, and online updates of $G=(V,E)$13 (Kato et al., 7 Aug 2025). This suggests that A-CMTS is best understood as a high-level congestion-aware routing primitive within a larger autonomy stack, rather than as a complete multi-agent execution framework.