---
title: 'A-CMTS: Anytime Congestion Mitigation Tree Search'
url: https://www.emergentmind.com/topics/a-cmts
type: topic
---

# A-CMTS: Anytime Congestion Mitigation Tree Search

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 [2508.05253]. 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 [2508.05253]. 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 $\omega \ge 1$ [2508.05253].

## 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 $s_i,g_i\in V$ [2508.05253]. 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 [2508.05253].

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

$$
f_e := |\{i\in A \mid \exists k, (\pi_i[k],\pi_i[k+1])=(u,v)\}|.
$$

The congestion degree at vertex $v$ is

$$
C(v) := \left(\prod_{e\in\delta^{-}(v)}(f_e+1)\right)-1,
$$

where $\delta^{-}(v)$ denotes the set of edges entering $v$ [2508.05253]. The objective is to minimize

$$
\sum_{v\in V} C(v).
$$

This multiplicative construction makes vertices where many agents enter from diverse directions become very expensive [2508.05253]. The illustrative example in the paper is deliberately extreme: if **20 agents all use one incoming edge**, then $C(v)=(20+1)-1=20$; if the same **20 agents are distributed equally over four incoming edges**, then $C(v)=6^4-1=1295$ [2508.05253]. 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 [2205.00955], bias single-agent search using predictive space-utilization heuristics [2109.04677], or reduce high-level cell congestion through hierarchical flow planning [2407.02777]. A-CMTS is distinguished by embedding the congestion penalty directly into a static routing objective on a sparse graph [2508.05253].

## 2. Optimization structure and exact formulation

The full CMPP model is a **mixed-integer nonlinear program**. Its decision variables are $z_{i,e}\in\{0,1\}$ for agent-edge usage, with auxiliary variables $f_e\in\mathbb{Z}^{0+}$ and $C(v)\in\mathbb{Z}^{0+}$ [2508.05253]. The optimization problem is

$$
\text{minimize}\quad \sum_{v\in V} C(v)
$$

subject to

$$
f_e = \sum_{i\in A} z_{i,e} \qquad \forall e\in E
$$

$$
C(v) = \prod_{e\in\delta^{-}(v)}(f_e+1) - 1 \qquad \forall v\in V
$$

$$
\sum_{e\in\delta^{+}(s_i)\cup\delta^{-}(g_i)} z_{i,e}=1 \qquad \forall i\in A
$$

$$
\sum_{e\in\delta^{-}(v)} z_{i,e} = \sum_{e\in\delta^{+}(v)} z_{i,e}
\qquad \forall i\in A,\; v\in V\setminus\{s_i,g_i\}
$$

$$
z_{i,(u,v)}+z_{i,(v,u)}\le 1
\qquad \forall i\in A,\; (u,v)\in E.
$$

The critical source of difficulty is the constraint

$$
C(v)=\prod_{e\in\delta^{-}(v)}(f_e+1)-1,
$$

which makes the problem **nonconvex** [2508.05253]. The paper implements an exact MINLP solver in **SCIP**, using **branch-and-bound over $z_{i,e}$ with nonlinear constraints enforced via outer approximation** [2508.05253]. Under a **60 s limit**, this solver succeeds on small instances but does not scale to larger ones. On a **$3\times 3$ 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 **$10\times 10$ Grid**, the solver **Timeout**s at **100** and **200** agents, and **N/A** is reported for **300** agents [2508.05253].

A-CMTS is introduced precisely against this backdrop: it is the scalable alternative to the exact MINLP, retaining a formal approximation guarantee through $\omega$ while avoiding direct global solution of the nonconvex program [2508.05253].

## 3. Two-layer search architecture

A-CMTS is described as **a two-layer search akin to CBS** [2508.05253]. Its high-level search maintains nodes $N$ storing

- $C^{+}$: forced $(\text{agent},\text{edge})$ constraints,
- $C^{-}$: forbidden constraints,
- $\pi$: current set of paths,
- $\text{cost}=\sum_v C(v)$,
- $\text{LB}$: a lower bound satisfying
  $$
  \text{LB} \le \min\left\{\sum_v C(v)\mid \pi \text{ satisfies } N.C^{+},N.C^{-}\right\}.
  $$

The high-level algorithm initializes the root node with empty $C^{+}$ and $C^{-}$, computes the root solution using **Prioritized Planning (PP) via low-level**, and inserts the node into a priority queue **Open ordered by cost** [2508.05253]. The incumbent solution is tracked through $\text{UB}$ and $\pi_{\text{Best}}$.

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

$$
v^* \in \Gamma(N) \text{ with highest } C(v),
$$

where

$$
\Gamma(N):=\{v \mid \exists a\in A,\; e\in\delta^{-}(v)\cap \pi_a \text{ s.t. } (a,e)\notin C^{+}\}.
$$

It then selects an agent $a^*$ and an incoming edge $e^*$ satisfying

$$
a^*\in\{a\mid \exists e^*\in\delta^{-}(v^*)\cap\pi_a,\; (a,e^*)\notin C^{+}\},
$$

and expands node $N$ into two children [2508.05253]. Child $P$ adds $(a,e)$ to $C^{+}$ and leaves the paths unchanged; child $Q$ adds $(a,e)$ to $C^{-}$, then replans $\pi_a$ via the low-level and also replans $\pi_{a'}$ for other agents $a'$ that visit $v$ [2508.05253]. 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 $\omega\ge 1$ controls pruning: if

$$
\text{UB}\le \omega \cdot N.\text{LB},
$$

node $N$ is pruned [2508.05253]. This produces the characteristic trade-off noted in the paper: **higher $\omega$ $\rightarrow$ more pruning $\rightarrow$ faster but looser bound** [2508.05253].

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 $C(v)$ values rather than spacetime collisions.

## 4. Low-level search and approximation guarantee

The low-level problem for a single agent $a$ under per-agent constraints $C_a^{+}\subseteq C^{+}$ and $C_a^{-}\subseteq C^{-}$ is to find $\pi_a$ minimizing **incremental congestion**:

$$
\text{minimize}\quad \sum_{(u,v)\in \pi_a}\Delta C(v)
$$

subject to $\pi_a[1]=s_a$, $\pi_a[L_a]=g_a$, $(u,v)\notin C_a^{-}$, and all $(u,v)\in C_a^{+}$ must appear in $\pi_a$ [2508.05253]. The implementation uses an approximation in which forced edges are visited **in order of Euclidean distance**, and **Dijkstra on $\Delta C(v)$** provides the route between them [2508.05253].

The solver’s formal guarantee is stated explicitly: **A-CMTS returns solution cost $c\le \omega\cdot c^*$** [2508.05253]. The proof sketch given is that any branch excluding the optimal path is pruned only when

$$
\text{UB}\le \omega\cdot \text{LB}\le \omega\cdot c^*.
$$

Setting $\omega=1$ ensures optimality [2508.05253]. 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** [2508.05253]. Compared with **min-cost flow**, where costs are **additive on edges, convex**, CMPP uses **multiplicative vertex costs across incoming edges, nonconvex** [2508.05253]. Compared with **online collision avoidance** such as ORCA/RVO and PIBT, A-CMTS supplies **global guidance** while leaving **fine-scale conflicts to reactive controllers** [2508.05253]. This division of labor is conceptually close to hierarchical congestion-aware frameworks in large-scale replanning [2407.02777], 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 $G$ [2508.05253]. 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 $\pi_i$ is computed, a waypoint queue

$$
Q_i=(\pi_i[2],\ldots,\pi_i[L_i-1],g_i^*)
$$

is formed, and ORCA steers the agent toward the head of $Q_i$; when the agent comes within threshold $r=5\,\text{m}$, that waypoint is popped and the agent proceeds to the next one [2508.05253].

For **discrete** deployment, the paper integrates with **PIBT** in **lifelong-MAPF** settings. A sparse graph $G$ is constructed by sampling grid vertices $V^*\to V$. Each agent’s next waypoint is the representative grid cell $g(\pi_i[2])$, or the goal if $L_i<3$. PIBT then plans one-step collision-free moves toward that waypoint. Agent positions on the sparse graph are updated via $f:V^*\to V$, and the visited vertex is removed from $\pi_i$, which can be reused as the initial solution for the next CMPP solve [2508.05253].

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 [2205.00955]; swarm planning through Virtual Tubes similarly uses centralized path generation and decentralized control inside the tube [2404.09200]. A-CMTS occupies the same systems layer, but with a vertex-penalty objective designed for dense graph-based traffic [2508.05253].

## 6. Empirical performance, scope, and limitations

The paper reports that A-CMTS scales empirically to **$|V|$ up to 2,500** and **$|A|$ up to 10,000 within seconds** [2508.05253]. This is the central practical result: the solver is intended for regimes far beyond what the exact MINLP can handle.

On the **$3\times 3$ Grid** with **$|A|=30$**, **A-CMTS ($\omega=1.0$)** improves the objective from **init cost = 90.7** to **final = 74.4**, an **18.0% improvement**, within **60 s** [2508.05253]. On the **$10\times 10$ Grid** with **$|V|=100$**, where the MINLP times out at **$|A|\ge 100$**, A-CMTS achieves **100% success**, produces **initial solutions in <0.1 s**, and yields **final solutions ~7% better** [2508.05253]. On **lak303d** with **$|V|=265$**, where the MINLP fails beyond **$|A|=200$**, A-CMTS yields **~9% cost reduction over initial** [2508.05253].

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 **$22\times 18.5\,\text{m}$** with **agent radius $0.3\,\text{m}$** and **horizon $60\,\text{s}$**, both vanilla ORCA and CMPP-guided ORCA achieve **100% success by 40 s** at **$|A|=200$**; at **$|A|=300$**, vanilla ORCA reaches **100% at ~60 s** while CMPP-guided ORCA reaches **100% at ~40 s**; at **$|A|=400$**, vanilla ORCA attains **83.9% at 60 s**, whereas CMPP-guided ORCA reaches **99.0%**, a **gain 15.1 pp** [2508.05253]. In **discrete lifelong-MAPF**, the reported throughput gains include **+58.1%** on **warehouse-10-20-10-2-1** with **$|A|=1{,}500$** and **+15.7%** on **random-64-64-10** with **$|A|=1{,}900$**, while **lak303d** shows a slight reduction due to **unavoidable single-lane bottlenecks** [2508.05253].

Runtime allocation also matters. Increasing the **A-CMTS runtime limit $t_L$ from $0.1\,\text{s}$ to $1\,\text{s}$** improves PIBT throughput by **up to 10%** [2508.05253]. 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** [2508.05253]. The cited future directions are to **integrate dynamic task allocation, product-placement optimization, robustness to unpredictable agent behaviors, and online updates of $G$** [2508.05253]. 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.

Source: https://www.emergentmind.com/topics/a-cmts