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Exploiting Subgraph Structure in Multi-Robot Path Planning
Malcolm Ross Kinsella Ryan
TL;DR
Multi-robot path planning becomes intractable as centralized state spaces grow, while alternative prioritization sacrifices completeness. The paper partitions maps into structured subgraphs, searches compact configuration spaces, and resolves abstract plans deterministically. The abstraction preserves soundness and completeness for centralized search, reduces search burden in structured maps, and complements prioritized planning despite its limitations.
Problem
Multi-robot planning must coordinate robots in shared environments, but centralized search scales poorly and decoupled or prioritized methods can be incomplete.
Method
The paper partitions road-maps into structured subgraphs, represents equivalent robot arrangements as configurations, and combines local planners hierarchically.
Results
The abstraction is sound and complete for centralized planning, supports deterministic plan resolution, and is most effective on sparsely connected realistic maps.
Takeaways & Limitations
Subgraph structure can make multi-robot planning much faster by compressing robot movements into abstract steps, with practical gains outweighing increased branching in the reported experiments.
Takeaways & Limitations
Prioritized planning remains incomplete, and some subgraph transitions can generate many possible configurations.
Abstract
from arXiv · showhide
Multi-robot path planning is difficult due to the combinatorial explosion of the search space with every new robot added. Complete search of the combined state-space soon becomes intractable. In this paper we present a novel form of abstraction that allows us to plan much more efficiently. The key to this abstraction is the partitioning of the map into subgraphs of known structure with entry and exit restrictions which we can represent compactly. Planning then becomes a search in the much smaller space of subgraph configurations. Once an abstract plan is found, it can be quickly resolved into a correct (but possibly sub-optimal) concrete plan without the need for further search. We prove that this technique is sound and complete and demonstrate its practical effectiveness on a real map. A contending solution, prioritised planning, is also evaluated and shown to have similar performance albeit at the cost of completeness. The two approaches are not necessarily conflicting; we demonstrate how they can be combined into a single algorithm which outperforms either approach alone.
1. Introduction
Multi-robot planning must coordinate independent goals in shared environments, but centralized search scales poorly while decoupled methods sacrifice completeness. The paper abstracts map structure into constrained subgraphs and combines local planners hierarchically.
- Centralized methods scale poorly as the number of robots increases, while decoupled methods are faster but incomplete.Decoupled planning can fail when robots must detour from optimal paths to let others pass.
- Road-map structures such as corridors and open spaces impose predictable constraints on robot interactions.Corridors can enforce first-in/first-out passage, whereas open spaces may allow simultaneous collision-free traversal.
- The paper combines local subgraph planners into a hierarchical planner for solving arbitrary problems.Planning first determines movements between subgraphs, then constructs paths within each subgraph.
- The proposed abstraction identifies subgraph types with known passage-order constraints and builds efficient local planners for them.The approach targets homogeneous robots navigating a shared road-map.
- With centralized search, the abstraction is sound and complete, while prioritized search retains its incompleteness but may find otherwise unavailable plans.Experiments find the approach most effective on maps with sparsely connected graph representations.
2. Problem Formulation
The problem is modeled as homogeneous robots moving on a shared graph-based road-map under localized collision assumptions. A centralized planner searches robot moves recursively, producing a composite state space that becomes expensive as map and robot counts grow.
- The road-map is a graph G=(V,E), and homogeneous robots share this graph with all starts and goals located on it.The graph represents free-space connectivity for a single robot, such as a cell decomposition or visibility graph.
- Collisions are assumed to occur only when a robot enters a vertex occupied, entered, or exited by another robot.Robots at other vertices do not affect the movement under this model.
- The centralized planner repeatedly chooses a robot and neighboring destination, moving it when the destination is unoccupied and backtracking on failure.The algorithm recursively explores robot and edge choices until all robots reach their goals.
- The centralized algorithm searches the composite space G^k formed by combining the graph positions of k robots.Collision states are removed from this composite graph.
- The composite graph can become expensive because its size depends on n=|V(G)| and k=|R|, with long runtimes expected for moderately large values.The paper’s formulation makes the dependence on both map size and robot count explicit.
3. Subgraph Abstraction
Subgraph abstraction treats arrangements with equivalent internal robot orderings as the same configuration, shrinking the planning space. Abstract transitions can then be resolved into concrete movements without further search.
- A small map with 18 vertices, 17 edges, and 3 robots yields a composite graph with 4896 vertices and 12240 edges.The example motivates grouping many robot arrangements that differ in exact positions but share relevant ordering.
- Stacks: A stack constrains robots to last-in-first-out passage, so internal order cannot change without exiting and re-entering.Reversing the order requires moving all robots out and then bringing them back in the opposite order.
- Stacks: For a stack, the configuration records robot order rather than each robot’s exact vertex position.Once the order is correct, rearranging robots into exact positions is trivial.
- Abstract state space: In the three-stack example, the abstract state space has only 60 states and 144 possible transitions.Transitions move the top-most robot from one stack to another, making the abstract space dramatically smaller than the composite space.
- Subgraph properties: Useful subgraphs allow transition computation from an abstract configuration rather than the exact internal arrangement.The paper later considers cliques, halls, and rings alongside stacks.
- Configuration space: If two arrangements share a configuration, transforming between them can be done easily without search, so planning need only search the smaller configuration space.This is the central reduction supplied by the abstraction.
4. Definitions
The formalism partitions the road-map into induced subgraphs and groups equivalent robot arrangements into configurations. Abstract transitions update these configurations, and abstract plans correspond to concrete plans.
- The map graph G is partitioned into disjoint induced subgraphs S1,...,Sm.An induced subgraph preserves exactly the edges between its vertices that exist in G.
- An arrangement is a one-to-one partial function from graph vertices to robots, specifying each robot’s location.The same notion applies to arrangements within an individual subgraph.
- A subgraph configuration groups arrangements that can transform into one another without robots leaving the subgraph.The whole-map configuration is represented as a tuple of subgraph configurations.
- The operators ⊕ and ⊖ represent robots entering and leaving subgraphs, with updates depending on the traversed edge.For complex subgraphs, these operators return sets because a transition may yield multiple configurations or none.
- An abstract plan is a sequence of transitions with intermediate configurations, and each abstract plan has at least one corresponding concrete plan, and vice versa.The correspondence follows from configuration equivalence and requires concrete subgraph transitions to appear in the abstract plan.
5. Subgraph Planning
Subgraph planning searches abstract configurations and deterministically resolves a found abstract plan into concrete robot movements. Complete abstract search therefore yields a sound and complete planner without further resolution search.
- Abstract planning: The planner searches configurations of subgraphs, extending an abstract plan by moving robots between neighboring subgraphs when transitions are applicable.Entering a subgraph may produce multiple successor configurations, requiring a configuration choice at each transition.
- Concrete resolution: Resolution constructs concrete movement sequences for each abstract transition, including reaching the outgoing vertex and arranging the entered subgraph.The six planning and resolution functions cover exits, entrances, termination, and their corresponding concrete rearrangements.
- Soundness and completeness: A complete abstract search guarantees a plan whenever one exists, because every abstract plan corresponds to at least one concrete plan.The resolution phase is deterministic once the abstract plan has been found.
- Efficiency: Efficient planning depends on subgraph-local transition and resolution functions that avoid expensive search and depend only on robots within the relevant subgraph.This locality is enabled by carefully constraining the allowed subgraph structures.
6. Subgraph Structures
The paper selects recurring, structurally constrained subgraphs whose configurations compactly encode robot order and whose actions can be resolved efficiently. It presents stacks, halls, cliques, and rings, using least commitment to reduce clique branching.
- Design criteria: Useful abstractions should occur commonly, be easy to extract, cover substantial search space, support fast sound transitions, and resolve efficiently.These criteria guide the selection of subgraph types for road-map partitions.
- Canonical structures: The paper presents four subgraph types: stacks, halls, cliques, and rings.The analysis denotes subgraph size by n and its pre-action robot count by k.
- Stacks: Stacks are narrow dead-end corridors where robots cannot pass and therefore enter and leave in last-in-first-out order.A stack configuration is the robot ordering from its head downward; entering a non-full stack adds one robot in O(1) time.
- Halls: Halls generalize stacks into narrow chains with multiple entrances and exits, while preserving robot order because passing is impossible.Hall entry configurations depend on hall size, occupancy, and entry index; limited space can restrict insertion positions.
- Cliques: Clique planning can use one locked configuration that records only the locking robot and vertex, leaving other robot permutations unspecified until needed.This least-commitment representation can significantly reduce search branching, with look-ahead arranging robots before a required exit or termination.
- Resolution trade-offs: The deterministic resolution procedures may sacrifice path optimality for speed in clique and ring cases, although smarter resolution planners could improve it.Because resolution occurs only once, the paper states that this trade-off need not have a major overall effect.
7. Prioritised Planning
Prioritised planning reduces search by committing robots to a priority order, but this sacrifices completeness. Combining priorities with delayed subgraph resolution broadens feasible plans while retaining abstraction benefits, subject to partition quality.
- Prioritised planning: Prioritised planning builds each robot’s plan in fixed priority order while requiring later plans not to interfere with earlier ones.The approach prunes alternatives, making search faster when a solution remains within the retained space.
- Completeness trade-off: A prioritised planner can miss valid solutions when robots must temporarily detour, such as swapping positions by moving aside for one another.Committing either robot to its individually optimal one-step plan prevents the other from finding a non-interfering plan.
- Combined approach: Prioritised search and subgraph representation are orthogonal: robots can be planned consecutively in abstract space and resolved only after compatible plans exist for all robots.This combines prioritised planning with subgraph abstraction rather than treating them as competing methods.
- Benefits: Delaying concrete resolution avoids high-priority commitments that could hamper later robots and lets the combined planner cover more possible plans.With a suitable hall partition, the method solves a swap by planning the lower-priority robot’s abstract excursion while resolving the higher-priority robot’s movements as needed.
- Partition dependence: The combined approach depends on choosing an appropriate partition and can fail on problems where a wrong abstraction blocks a solution available to standard prioritised planning.Such cases can occur with larger numbers of robots.
8. Search Complexity
Subgraph decomposition changes search complexity by reducing plan depth while potentially increasing branching through graph connectivity and multiple resulting configurations. Its effectiveness therefore depends on careful partitioning.
- Search representation: Subgraph transitions act as macro-operators between abstract states defined by sets of configurations.They replace detailed movement sequences with higher-level transitions in the search.
- Search representation: Search cost depends on goal depth, average branching factor, and search efficiency.The search tree is shaped by d, b, and the efficiency of the search procedure.
- Trade-offs: Macro-operators usually reduce search depth but can increase branching, making them most beneficial when depth dominates.Their advantage is stronger in uninformed search and weaker when a good heuristic already balances depth and branching.
- Trade-offs: Subgraph branching increases when reduced-graph degree rises, with sparse structures such as halls worse than dense structures such as cliques.A subgraph can expose more outgoing edges than a single vertex, especially when those edges lead to different subgraphs.
- Trade-offs: A single transition can generate many configurations when a robot enters an occupied large hall, although least commitment can avoid unnecessary alternatives.Different generated configurations may still matter when they lead to different outcomes.
- Practical implication: Careful decomposition is important, but an appropriate partition can significantly improve both informed and uninformed search.The experiments identify decomposition choice as central to avoiding excessive branching.
9. Experiments
Across experiments, subgraph abstraction reduced search depth and often improved planning speed or completeness, especially for constrained problems, but its benefit depended on graph structure, heuristic quality, and decomposition quality.
- 9.1.1 Scaling |V|: Across vertex-scaling experiments, both complete and prioritised planners improved significantly over their naive alternatives when using the subgraph abstraction.The experiments still exhibited combinatorial runtime growth as graph size increased.
- 9.1.1 Scaling |V|: Naive prioritised search missed available solutions, whereas prioritised subgraph search avoided those failures in the reported experiments.Three failures occurred for naive prioritised search in Table 1; the prioritised subgraph search did not encounter this problem.
- 9.1.2 Scaling |E|: As graph density increased, runtime changed little overall, but prioritised subgraph search solved nearly all problems while naive prioritised search failed on up to 10% of sparse graphs.With the abstraction, only two problems remained unsolved in the density experiment, and none were solved by the naive planner alone.
- 9.1.3 Scaling |R|: With more robots, prioritised planners handled twice as many robots as complete planners, while subgraph abstraction reduced prioritised failures from 46% at 10 robots to three failures in 1,000 problems.The complete-search abstraction also gained a significant advantage as robot count increased, after overhead on very small problems.
- 9.1 Experiments: The subgraph abstraction reduced goal depth while increasing branching factor; under uninformed search, the depth reduction dominated and improved planning time.The authors describe uninformed search as having expected running time O(b^d), with depth exerting the stronger effect in these experiments.
- 9.2 Experiment 2: Heuristic Search: Under heuristic search on larger random graphs, subgraph approaches lost their advantage because branching-factor growth outweighed their goal-depth reduction.The authors attribute this partly to poor auto-partitioning and to heuristics containing less information for subgraph configurations.
10. Conclusion
The paper presents subgraph-based abstraction as a faster, complete approach to multi-robot planning and shows that combining it with prioritisation is more powerful. It also identifies partitioning, symmetry exploitation, and richer subgraph structures as directions for further work.
- Conclusion: Subgraph decomposition enables faster multi-robot planning without sacrificing completeness by treating many robot arrangements as equivalent configurations.The paper describes stacks, halls, cliques, and rings as structures common enough for realistic maps and simple enough for efficient configuration computation.
- Conclusion: Abstract plans can be resolved deterministically into concrete plans without further search, although the resulting plans are not necessarily optimal.The authors suggest spending more time in resolution to trim wasteful paths.
- Conclusion: Combining subgraph-based planning with prioritisation is more powerful than either method alone and can partly alleviate prioritised planning's incompleteness.The paper compares subgraph-based planning favorably with prioritised planning while retaining the possibility of combining them.
- Related work: Earlier hierarchical path-planning abstractions do not carry over well to multi-robot scenarios, while this method makes no garage-location assumption.The authors characterize their approach as more general in application and report no previous complete abstraction-based planner for the general multi-robot problem.
- Future work: Future work includes exploiting subgraph symmetries, improving automatic partitioning, formalizing additional structures, and expressing the problem as a constraint satisfaction problem.The proposed partitioning work explicitly targets the trade-off between search depth and branching factor.
Appendix A. Proof of Soundness and Completeness
The appendix proves soundness and completeness of the abstract planning process by establishing equivalence between abstract-plan existence and concrete-plan existence.
- Proof result: An abstract plan exists for a given problem if and only if a concrete plan also exists.The appendix states this as the main theorem after setting up the formal definitions.
A.1 Graphs and Subgraphs
The appendix defines induced subgraphs and partitions, then contracts partition subgraphs into vertices of a reduced graph that represents connections between them.
- Induced subgraphs: An induced subgraph contains a subset of the parent graph's vertices and all edges connecting those vertices.Because all connecting edges are retained, the subgraph can be specified solely by its vertices.
- Partitions: A partition divides the graph into subgraphs whose vertex sets are pairwise disjoint.The partition is represented as a set of subgraphs S1, . . . , Sm.
- Reduced graph: The reduced graph X is constructed by contracting each partition subgraph to one vertex, with edges recording connections between distinct subgraphs.An edge exists when some vertex in one subgraph is connected to a vertex in another in the original graph.
A.2 Robots and Arrangements
The appendix represents robot locations as arrangements and shows that an arrangement can be uniquely reconstructed from its induced arrangements over a graph partition.
- Arrangements: An arrangement is a one-to-one partial function from graph vertices to robots, representing robot locations and allowing unoccupied vertices.It need not include every robot in the robot set.
- Local arrangements: An induced arrangement restricts a global arrangement to a particular subgraph, while combined arrangements merge disjoint local arrangements.The combination operator applies when the subgraphs and their arrangements are disjoint.
- Reconstruction: The combined induced arrangements over a partition reconstruct the original arrangement, uniquely identifying robot locations in the full graph.Lemma 1 states that combining all induced arrangements returns the original arrangement.
A.3 Concrete Plans
The paper abstracts robot arrangements into subgraph configurations and searches transitions between them, while preserving correspondence with concrete plans. This abstraction supports a sound and complete planning procedure whose abstract search is followed by deterministic resolution.
- A.4 Configurations: Configurations group arrangements that can be transformed into one another within a subgraph without robots leaving it.The resulting equivalence classes capture, for example, robot ordering in a stack subgraph.
- A.4 Configurations: A configuration tuple records the configuration of every partition subgraph and abstracts the entire graph’s robot arrangement.Concrete arrangements belong to a tuple when their restrictions belong to the corresponding subgraph configurations.
- A.5 Abstract Plans: Abstract plans consist of subgraph transitions between configuration tuples, moving robots across partition boundaries when departure and entry conditions are satisfied.A transition is applicable when the source can arrange the robot at its outgoing vertex and the destination can make its incoming vertex available.
- A.5 Abstract Plans: An abstract plan exists exactly when a corresponding concrete plan exists, establishing equivalence between the reduced search space and the original planning problem.The theorem supports searching abstract plans instead of all concrete movement sequences.