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Solving Multi-Agent Multi-Goal Path Finding Problems in Polynomial Time

Stefan Edelkamp

TL;DR

The paper addresses multi-agent multi-goal path finding on undirected graphs by introducing polynomial-time solutions for Anonymous MAPF and Combinatorial MAPF. It fuses shortest-path reductions, assignment optimization via the Hungarian algorithm, and a conflict-resolution mechanism inspired by ants-on-a-stick to keep agents on precomputed shortest paths while dynamically reassigning goals at collisions. For CMAPF, it leverages NRPA rollouts to generate tours and then applies on-the-fly conflict resolution, yielding practical, near-optimal results in many cases. Experiments on 500 benchmark instances and large-scale maps show substantial reductions in makespan, sum-of-cost, and conflicts with practical runtimes, highlighting the approach's scalability for dense, multi-goal tasks.

Abstract

In this paper, we plan missions for a fleet of agents in undirected graphs, such as grids, with multiple goals. In contrast to regular multi-agent path-finding, the solver finds and updates the assignment of goals to the agents on its own. In the continuous case for a point agent with motions in the Euclidean plane, the problem can be solved arbitrarily close to optimal. For discrete variants that incur node and edge conflicts, we show that it can be solved in polynomial time, which is unexpected, since traditional vehicle routing on general graphs is NP-hard. We implement a corresponding planner that finds conflict-free optimized routes for the agents. Global assignment strategies greatly reduce the number of conflicts, with the remaining ones resolved by elaborating on the concept of ants-on-the-stick, by solving local assignment problems, by interleaving agent paths, and by kicking agents that have already arrived out of their destinations

Solving Multi-Agent Multi-Goal Path Finding Problems in Polynomial Time

TL;DR

The paper addresses multi-agent multi-goal path finding on undirected graphs by introducing polynomial-time solutions for Anonymous MAPF and Combinatorial MAPF. It fuses shortest-path reductions, assignment optimization via the Hungarian algorithm, and a conflict-resolution mechanism inspired by ants-on-a-stick to keep agents on precomputed shortest paths while dynamically reassigning goals at collisions. For CMAPF, it leverages NRPA rollouts to generate tours and then applies on-the-fly conflict resolution, yielding practical, near-optimal results in many cases. Experiments on 500 benchmark instances and large-scale maps show substantial reductions in makespan, sum-of-cost, and conflicts with practical runtimes, highlighting the approach's scalability for dense, multi-goal tasks.

Abstract

In this paper, we plan missions for a fleet of agents in undirected graphs, such as grids, with multiple goals. In contrast to regular multi-agent path-finding, the solver finds and updates the assignment of goals to the agents on its own. In the continuous case for a point agent with motions in the Euclidean plane, the problem can be solved arbitrarily close to optimal. For discrete variants that incur node and edge conflicts, we show that it can be solved in polynomial time, which is unexpected, since traditional vehicle routing on general graphs is NP-hard. We implement a corresponding planner that finds conflict-free optimized routes for the agents. Global assignment strategies greatly reduce the number of conflicts, with the remaining ones resolved by elaborating on the concept of ants-on-the-stick, by solving local assignment problems, by interleaving agent paths, and by kicking agents that have already arrived out of their destinations
Paper Structure (10 sections, 4 theorems, 17 figures, 1 table)

This paper contains 10 sections, 4 theorems, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Solution Principle
  4. Polynomial-Time Solutions
  5. Anonymous MAPF
  6. Combinatorial MAPF
  7. Experiments
  8. Anonymous MAPF
  9. Combinatorial MAPF
  10. Conclusion

Key Result

Theorem 1

For any given tour assignment of $k$ infinitely small agents traveling from their starting locations to a set of goals in linear motion along the shortest-path edges in an undirected graph embedded in the Euclidean plane, let the total (maximal) travel time $L$ correspond to the sum (max) of the tra

Figures (17)

  • Figure 1: Regular, Anonymous, and Combinatorial MAPF (left to right). where the search graph nodes are line intersections. red stars represent target nodes, obstacles are shown as gray circles, agents are colored dots and their travel is indicated by a chain of colored edges of the agent's color. For MAPF a fixed assignment from goals to agents is enforced, in anonymous MAPF there are as many goals as agents, while for CMAPF there might be more goals than agents.
  • Figure 2: Ants-on-a-stick problem, exchanging direction on collision with one ant falling off at and of stick.
  • Figure 3: Agents-on-a-graph problem with agents following their individual shortest-paths, while exchanging goal agendas (that are attached to the agents) on collision.
  • Figure 4: Cuckoo'ing: pushing agents that have arrived at goal from their place. Agent 1 will continue with purple shortest path of Agent 2. This procedure adds shortcuts to the agent dependency graph and the interaction has to be implemented with care.
  • Figure 5: Zipping: For node conflicts of agents pointing in the same direction the agents of one agent dependency subtree have to wait. The other branches are dealt with in next iteration.
  • ...and 12 more figures

Theorems & Definitions (6)

  • Theorem 1: Polytime Constant-Approximation
  • Definition 1: Discrete Anonymous MAPF
  • Theorem 2: Polytime AMAPF
  • Theorem 3: Polytime Single-Agent CMAPF
  • Definition 2: Discrete Combinatorial MAPF
  • Theorem 4: Polynomial-Time CMAPF