MAPF-Collapse: Optimization & Deadlock Resolution

• MAPF-Collapse is a formal problem in multi-agent path finding that minimizes redundant movements through trajectory-level collapse operations.
• It is NP-hard, with formulations like Judgelight using ILP to optimize path cost while preserving collision-free constraints.
• The framework also addresses behavioral deadlocks in learning-based controllers by integrating hybrid RL with explicit deadlock detection for robust navigation.

MAPF-Collapse denotes a critical phenomenon and formal post-optimization problem in Multi-Agent Path Finding (MAPF), with two principal usages in recent literature: (1) the identification and elimination of topologically redundant or oscillatory movement in existing multi-agent schedules via trajectory-level post-processing; and (2) the emergence of persistent deadlocks—"collapse" in collective agent progress—particularly in learned or fully decentralized controllers. The first notion is formalized as an NP-hard combinatorial problem ("MAPF-Collapse") and addressed by the Judgelight framework (Tang et al., 27 Jan 2026); the second denotes behavioral collapse that motivates hybrid architectures with explicit deadlock detection (Wang et al., 27 Nov 2025). Both are central to attaining real-world deployability and performance guarantees in algorithmic multi-robot coordination.

1. Formal Problem Statement and Collapse Operation

In the context of MAPF on an undirected graph $G=(V,E)$, a feasible schedule $M$ specifies, for $N$ agents over a time horizon $T$, collision-free, edge-adjacent trajectories $M\in V^{N \times (T+1)}$ from respective starts to goals. A closed subwalk in an agent's trajectory $[v^i_a, \ldots, v^i_b]$ is a segment with $v^i_a = v^i_b =: x$. The elementary collapse operation replaces all per-step entries $v^i_k$ in $[a,b]$ by $x$, effectively turning an excursion that leaves and returns to $x$ into a sequence of waits at $x$. The cost metric of interest is flowtime: the number of non-self-loop, i.e., move, actions across all agents,

$$\mathrm{cost}(M) = \sum_{i=1}^N \left| \{ t \mid v^i_t \neq v^i_{t+1} \} \right|.$$

The MAPF-Collapse optimization seeks, given a feasible $M$, a sequence of valid collapses yielding a new schedule $M'$ of minimal cost, while preserving all start, goal, edge, and collision constraints (Tang et al., 27 Jan 2026).

2. Computational Complexity

The decision variant, $\mathrm{DMAPF\text{-}Collapse}(G, M, \beta)$, asks if a feasible, collapsed schedule $M'$ of cost at most $\beta$ exists. Membership in NP is established via $O(NT)$-time verification of edge-adjacency and collision-freedom for any proposed $M'$ (Lemma 1). NP-hardness follows by reduction from Independent Set, using a construction in which savings achievable by collapse correspond directly to large independent sets in the underlying graph: specific arrangements of "vertex-agents" and "edge-agents" encode independent-set cardinality as collapse-induced cost reduction. The transformation certifies NP-completeness for the decision and NP-hardness for the optimization variant (Theorem 1 in (Tang et al., 27 Jan 2026)).

3. Integer Linear Programming Formulation

Judgelight formalizes MAPF-Collapse as an ILP over a set $\mathcal{C}$ of candidate collapses $c=(i,a,b,x)$. For each $c$, introduce a boolean $y_c$, weighted by the number of edge-traversals $w_c$ eliminated if $c$ is applied:

$$w_c = \sum_{k=a}^{b-1} \mathbf{1}[ M[i,k] \neq M[i,k+1] ].$$

The ILP maximizes $\sum_c w_c y_c$, subject to four key classes of constraints:

1. Intra-agent mutual exclusion: no overlapping or nested intervals for a single agent,
2. Cross-agent exclusive occupancy: no two agents can collapse to the same vertex during overlapping intervals,
3. Collision-avoidance dependencies: if one agent collapses to $x$ at $k$, any agent occupying $x$ at $k$ must simultaneously collapse away,
4. Action invalidation: infeasible actions are forced to $y_c=0$.

This ILP contrasts with classical TSP in that no explicit subtour-elimination is necessary; correctness derives from the mutual-exclusion and cross-agent constraints. Candidate pruning and preprocessing (e.g. merging "ABA" oscillations) drastically reduce variable and constraint counts, enabling practical optimization for moderately large instances with state-of-the-art solvers such as Gurobi 13.0 (Tang et al., 27 Jan 2026).

4. Empirical Evaluation and Impact

Judgelight's MAPF-Collapse module was evaluated on 3,296 POGEMA benchmarks spanning diverse map typologies (Maze, Random, Warehouse, Puzzle, Cities-Tiles) and compatible with both learning-based (SCRIMP, DCC, RAILGUN, MAMBA, Follower) and search-based (LaCAM) solvers. Key metrics include total Solution Cost (SoC), SoC saving ratio, and individual success rate (ISR). Across all learning-based solvers, post-optimization consistently yields 20–40% SoC savings, with maximal benefit in high-density, "hard" instances; even high-quality search-based solvers see ≈10% improvement. In over 90% of test cases, runtime is under 1 s, and the fastest cases (e.g. LaCAM at high density) complete post-optimization in <100 ms. This demonstrates that MAPF-Collapse not only reduces route redundancy but does so with computational efficiency compatible with real-time or near-real-time applications (Tang et al., 27 Jan 2026).

5. MAPF-Collapse in RL and Hybrid Architectures

MAPF-Collapse also denotes a behavioral pathology in learning-based and decentralized MAPF, whereby agents become persistently deadlocked, especially in narrow or congested topologies. Reciprocal velocity obstacles (VO/RVO/ORCA) and RL controllers, while responsive, lack the topological awareness to guarantee escape from such "collapse" scenarios. Hybrid frameworks address this by deploying a continuous deadlock detector layered atop an RL policy, triggering an on-demand, locally confined MAPF solver (e.g., Push-and-Rotate) precisely when collapse is detected. This guarantees topological progress in polynomial time whenever the local environment (e.g., number of blanks in the grid) permits. The RL policy seamlessly resumes control after deadlock resolution, maintaining reward shaping and observation encoding throughout (Wang et al., 27 Nov 2025).

6. Theoretical Guarantees and Practical Considerations

On-demand MAPF-correction ensures two formal properties for implicated regions and agents: (1) deadlocks are cleared in finite steps whenever the local MAPF instance is solvable (i.e., sufficient "blanks" are available), and (2) the computational overhead of each intervention is polynomial in region size and agent count. Empirically, Table 1 in (Wang et al., 27 Nov 2025) shows hybrid RL+MAPF approaches raise multi-agent navigation success rates in canonical bottlenecks (Doorway, Corridor) from as low as 0–48% (pure RL) to >95–100%, effectively eliminating persistent collapse in unseen, dense layouts.

7. Significance and Broader Context

MAPF-Collapse, in both its post-optimization and deadlock senses, addresses critical bottlenecks in scalable, robust multi-agent coordination. The trajectory-level perspective enables aggressive schedule pruning for cost-efficiency and deployment-readiness, while the behavioral perspective motivates architectural designs with explicit completeness guarantees. These frameworks bridge the gap between fast-but-myopic learning-based navigation and slower but topologically complete search-based planning, yielding systems that attain both real-time performance and theoretical robustness in complex, real-world domains (Tang et al., 27 Jan 2026, Wang et al., 27 Nov 2025).