---
title: Dynamic Agent Grouping ECBS (DAG-ECBS)
url: https://www.emergentmind.com/topics/dynamic-agent-grouping-ecbs-dag-ecbs
type: topic
---

# Dynamic Agent Grouping ECBS (DAG-ECBS)

Dynamic Agent Grouping ECBS (DAG-ECBS) is a bounded-suboptimal windowed Multi-Agent Path Finding (MAPF) method designed for the Windowed Complete MAPF (WinC-MAPF) framework. Its central contribution is to show that partial-horizon planning can retain completeness even when the action generator is bounded-suboptimal rather than optimal, provided that planning is carried out over dynamically formed disjoint agent groups and that each group solution is \(w\)-bounded suboptimal. In this construction, dynamic grouping is not merely a search-efficiency device: it is the mechanism that makes the WinC-MAPF heuristic argument go through under bounded suboptimality [2509.15381].

## 1. Problem setting and windowed MAPF formulation

DAG-ECBS is formulated in the standard discrete-time MAPF model. There are \(N\) agents with start and goal locations \(C_i^{\text{start}}\) and \(C_i^{\text{goal}}\), and a joint configuration at time \(t\) is
\[
C^t = [C_1^t, C_2^t, \dots, C_N^t].
\]
A valid solution is a collision-free sequence
\[
C^{0:T} = \{C^0,\dots,C^T\}
\]
satisfying the start and goal conditions, obstacle avoidance, and the standard prohibition of vertex and edge collisions. The transition cost is unit-cost per move, except that waiting at the goal is free, and the objective is the sum of costs
\[
\sum_{i=1}^N \sum_{t=0}^{T-1} c(C_i^t,C_i^{t+1}).
\]

The method is explicitly designed for **windowed** or **partial-horizon** MAPF. Instead of solving to the final goals at every iteration, it plans only the next \(W\) timesteps and evaluates the terminal state using a backward-Dijkstra heuristic:
\[
\sum_{i=1}^N \left( \sum_{t=0}^{W-1} c(C_i^t,C_i^{t+1}) + h_i^{BD}(C_i^W)\right).
\]
This is the same setting in which WinC-MAPF introduced completeness-preserving heuristic learning for partial-path planning. The difficulty is that a myopic planner can enter deadlock or livelock: locally sensible \(W\)-step plans need not imply eventual progress. DAG-ECBS addresses exactly this gap by providing a bounded-suboptimal action generator compatible with WinC-MAPF’s completeness machinery [2509.15381].

Within WinC-MAPF, an action generator returns a terminal window configuration
\[
\arg\min_{C^W} c(C^0,C^W) + h(C^W),
\]
after which the heuristic at the current configuration is updated by
\[
h(C^0) \gets U(C^0,C^W) := \max(h(C^0), c(C^0,C^W) + h(C^W)).
\]
The original WinC-MAPF proof required this action generator to be optimal. DAG-ECBS replaces that requirement with a weaker but structured condition: \(w\)-bounded suboptimality must hold **per disjoint group**, not only globally.

## 2. Dynamic agent grouping as the core planning mechanism

DAG-ECBS begins from singleton groups,
\[
\text{activeGroups} \gets \{\{a\}\mid a\in\mathcal{A}\},
\]
plans each group independently, and merges groups only when their tentative windowed plans interact. The output is therefore not just a windowed trajectory \(C^{1:W}\), but also a set of **disjoint agent groups** that support additive heuristic reasoning in WinC-MAPF [2509.15381].

Algorithmically, the workflow is iterative. A group \(Gr\) is taken from `activeGroups` and solved by `Group-ECBS`, producing a bounded-suboptimal windowed plan \(\hat C_{Gr}^{1:W}\). This tentative plan is then checked against already planned groups. Any group whose current plan conflicts with \(\hat C_{Gr}^{1:W}\) is collected into
\[
Gr_{int} \gets \{Gr' \mid C^{1:W}_{Gr'} \text{ conflicts with } \hat C^{1:W}_{Gr}\}.
\]
If there are no interacting groups, \(Gr\) is accepted into `disjointGroups`. If there are conflicts, \(Gr\) and all groups in \(Gr_{int}\) are merged, removed from their previous status, and reinserted into `activeGroups` for replanning as a larger coupled unit. If `Group-ECBS` fails, DAG-ECBS returns failure.

The relevant notion of conflict is broader than ordinary path intersection. The paper includes three types of interactions: standard **vertex conflicts**, standard **edge conflicts**, and **heuristic conflicts**. This last category is inherited from the SS-CBS/WinC-MAPF setting, where learned heuristic penalties must be represented explicitly in the planner rather than injected as a naïve additive term.

This grouping rule is dynamic in two senses. First, it is recomputed at each planning iteration rather than fixed offline. Second, agents may be separate in one iteration and merged in another, depending on whether their current windowed plans interact. The resulting groups differ from the meta-agent constructions of MA-CBS or NECBS: here the point is not primarily to reduce repeated conflict resolution overhead, but to ensure that the final disjoint groups support the per-group boundedness property required by the completeness proof [2509.15381].

## 3. Modified ECBS search and heuristic machinery

DAG-ECBS uses a modified ECBS inside each group. The paper states that `Group-ECBS` is “nearly identical” to regular windowed ECBS, but with two essential changes: it incorporates heuristic penalties and heuristic conflicts as in SS-CBS, and it modifies focal-search criteria so that learned heuristics remain \(w\)-admissible rather than strictly admissible [2509.15381].

The heuristic is initialized as
\[
h(C) = w \cdot h^{BD}(C),
\]
and decomposed as
\[
h(C) = h_p(C) + w \cdot h^{BD}(C),
\]
where \(h_p\) is the learned penalty term. This differs from the original WinC-MAPF decomposition, which used \(h^{BD}(C) + \sum_i h_p(C_{Gr_i})\) under ordinary admissibility. In DAG-ECBS, the target invariant is
\[
h(C_{Gr}) \le w \cdot h^*(C_{Gr})
\]
for every disjoint group \(Gr\).

At the low level, regular ECBS uses the anchor value
\[
F_1(v|n) := c(C_i^0,v|n) + h_i^{BD}(v),
\]
with focal eligibility based on \(c+h\). DAG-ECBS changes the focal condition to
\[
\{v \mid c(C_i^0,v|n) + w\cdot h_i^{BD}(v) \le w \cdot \min_{v \in \text{Anchor}} F_1(v|n)\}.
\]
The structural change is that the heuristic term on the left-hand side is explicitly multiplied by \(w\). This is not a cosmetic modification: it is the search-level expression of the shift from admissibility to \(w\)-admissibility.

At the high level, regular ECBS uses
\[
F_2(n) := \sum_i \min_{v \in \text{Anchor}_i} F_1(v|n).
\]
DAG-ECBS instead defines
\[
F_3(n) := h_p(n) + w \cdot \sum_i \min_{v \in \text{Anchor}_i} F_1(v|n),
\]
and its focal condition becomes
\[
\{ n' \mid c(C^0,C^W) + w \cdot h^{BD}(C^W) + h_p(n) \le \min_{n \in \text{Anchor}} F_3(n) \}.
\]
The heuristic penalty \(h_p\) is therefore part of the high-level search state, and heuristic conflicts must be resolved analogously to path conflicts.

DAG-ECBS also updates not only the current configuration but intermediate states along the returned group path. For \(t\in[1,W-1]\), the appendix gives
\[
h(C^t_{Gr}) \gets \max\left(h(C^t_{Gr}),  U(C^0_{Gr},C^W_{Gr}) - w \cdot c(C^0_{Gr},C^t_{Gr})\right).
\]
This preserves the same \(w\)-admissibility invariant over intermediate window states.

## 4. Guarantees, proof structure, and common misunderstandings

The principal theorem is group-local:
\[
\text{For every disjoint agent group, the heuristic maintained by DAG-ECBS is always } w\text{-admissible, i.e., } h(C_{Gr}) \leq w \cdot h^*(C_{Gr}).
\]
This theorem is the key technical bridge from bounded-suboptimal search to WinC-MAPF completeness [2509.15381].

The proof has the standard WinC-MAPF / real-time heuristic-search shape. Heuristics are initialized in a \(w\)-admissible way. Assuming all current group heuristics are \(w\)-admissible, the modified anchor/focal structure of `Group-ECBS` ensures that the returned group transition satisfies
\[
c(C^0_{Gr},C^W_{Gr}) + h(C^W_{Gr}) \le w \cdot h^*(C^0_{Gr}).
\]
Because heuristic updates take the maximum of the old value and this bounded target, the updated heuristic remains \(w\)-admissible. In a finite state space, repeated replanning cannot cycle forever: if the system revisited the same local minima indefinitely, heuristic values would have to increase without bound, contradicting the \(w \cdot h^*\) upper bound.

A common misunderstanding is to assume that standard global ECBS bounded suboptimality is already sufficient. The paper explicitly shows why that is false. If Group 1 has optimal cost \(10\), Group 2 has optimal cost \(40\), and \(w=2\), then a total-cost solution of \(75\) is globally valid because \(75 \le 2\cdot 50\). But if the group costs are \(30\) and \(45\), Group 1 violates its local bound since \(30 > 2\cdot 10\). WinC-MAPF’s proof requires the local bound, not just the global one [2509.15381].

Another recurring confusion concerns the scope of the quality guarantee. DAG-ECBS does **not** claim a global end-to-end suboptimality bound for the entire executed trajectory over all replanning episodes. The bounded-suboptimality guarantee is local to each windowed planning call and to each final disjoint group. The completeness claim is about eventual reachability of the goals under repeated planning and heuristic updates, not about a global approximation factor for the full mission cost.

## 5. Empirical behavior and scaling properties

The evaluation compares DAG-ECBS against naive windowed ECBS and SS-CBS on benchmark maps from Stern et al. (2019). The reported parameters are
\[
W=\{1,2,4,8,16,32\}, \qquad w=\{1,1.5,2,3,5\},
\]
with a one-minute timeout summed across all planning iterations. The maps discussed explicitly in the analysis are `random-32-32-20`, `warehouse-10-20-10-2-1`, `ht_chantry`, and `den520d` [2509.15381].

The headline empirical conclusion is twofold. First, DAG-ECBS improves scalability compared to SS-CBS, showing that bounded-suboptimal planning can be made compatible with complete windowed MAPF. Second, DAG-ECBS can outperform naive windowed ECBS when completeness matters most, especially for small windows and in congested settings where deadlock and livelock are frequent. The paper identifies `random-32-32-20` and `warehouse-10-20-10-2-1` as maps where this advantage is strongest.

The reverse regime also appears. On `ht_chantry` and `den520d`, especially at larger window sizes, plain windowed ECBS can outperform DAG-ECBS. The interpretation given is not that the completeness machinery is ineffective, but that its overhead can dominate when local congestion is less severe and naive ECBS already keeps making progress. In those settings, DAG-ECBS can time out because it pays for group maintenance, heuristic penalties, heuristic conflict handling, and modified focal conditions.

The paper also reports several parameter trends. With fixed \(w=2\), increasing \(W\) increases per-iteration runtime for both methods, but appears to affect DAG-ECBS more strongly; many DAG-ECBS failures at large windows are runtime failures rather than deadlock failures. With fixed \(W=4\), increasing \(w\) consistently reduces DAG-ECBS per-iteration runtime. The authors attribute part of the difference from plain ECBS to the stricter DAG-ECBS focal condition: because \(w\cdot h_i^{BD}\) appears on the left-hand side, fewer nodes satisfy focal eligibility than in ordinary windowed ECBS. The text does not provide full numeric tables; scalability is described in terms of the maximum number of agents with success rate \(>50\%\), but the exact values remain in figures rather than the transcribed text [2509.15381].

## 6. Position within the ECBS literature

DAG-ECBS occupies a specific point in the ECBS design space. Standard ECBS is a globally bounded-suboptimal CBS variant; it is scalable, but naive windowed ECBS has no completeness guarantee in partial-horizon planning. SS-CBS, by contrast, is an optimal action generator for WinC-MAPF and is complete, but the paper presents it as less scalable in harder settings. DAG-ECBS combines the windowed-completeness framework of WinC-MAPF with bounded-suboptimal search by replacing global boundedness with per-group boundedness and by dynamically constructing disjoint groups [2509.15381].

This grouping role distinguishes DAG-ECBS from several other ECBS-family extensions. EECBS changes high-level node ranking and selection through Explicit Estimation Search while preserving the CBS/ECBS decomposition; it is therefore a search-control refinement rather than a grouping mechanism [2010.01367]. DECBS tightens lower bounds in focal search by replacing low-level focal search with a two-phase double search, again targeting search control rather than dynamic grouping [2503.03779]. CBS-Budget replaces ECBS-style low-level focal search with budgeted class-ordered search and a different high-level bound \(b(N)\), but still remains per-agent rather than group-dynamic in the DAG-ECBS sense [2206.00130].

Two additional contrasts clarify what DAG-ECBS is not. It is not an external spatio-temporal decomposition wrapper of the type used in “Spatial and Temporal Splitting Heuristics for Multi-Robot Motion Planning,” where ECBS is treated as a black-box solver for subinstances produced by space and time splitting; that approach localizes interactions by preprocessing the instance, not by dynamic grouping within the windowed search loop [2103.14111]. Nor is it an ECBS method for dynamic uncontrollable agents of the CP-Solver type, where the core novelty is prediction, conformal uncertainty, and asymmetric conflict handling, rather than grouping controllable agents into bounded-suboptimal disjoint sets [2507.22282].

A plausible implication, consistent with how the DAG-ECBS paper frames its broader significance, is that the method is best understood as a **blueprint** rather than an isolated algorithm. The paper explicitly identifies adaptation of more advanced bounded-suboptimal solvers—especially a future “DAG-EECBS”—as a natural next direction, and it also suggests decentralized variants in which nearby agents coordinate while distant agents plan independently [2509.15381]. In that sense, DAG-ECBS is both a concrete planner and a formal recipe for turning bounded-suboptimal MAPF solvers into complete windowed action generators.

Source: https://www.emergentmind.com/topics/dynamic-agent-grouping-ecbs-dag-ecbs