Multi-agent Pathfinding with n Agents on Graphs with n Vertices: Combinatorial Classification and Tight Algorithmic Bounds

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)


We investigate the multi-agent pathfinding (MAPF) problem with n agents on graphs with n vertices: Each agent has a unique start and goal vertex, with the objective of moving all agents in parallel movements to their goal s.t. each vertex and each edge may only be used by one agent at a time. We give a combinatorial classification of all graphs where this problem is solvable in general, including cases where the solvability depends on the initial agent placement.

Furthermore, we present an algorithm solving the MAPF problem in our setting, requiring \(\mathcal {O}(n^2)\) rounds, or \(\mathcal {O}(n^3)\) moves of individual agents. Complementing these results, we show that there are graphs where \(\Omega (n^2)\) rounds and \(\Omega (n^3)\) moves are required for any algorithm.


Permutation Group Full Version Combinatorial Condition Label Problem Agent Move 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank the anonymous reviewers for their helpful comments. Klaus-Tycho Foerster is supported by the Danish Villum Foundation.


  1. 1.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hart, P.E., Nilsson, N.J., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. 4(2), 100–107 (1968)CrossRefGoogle Scholar
  3. 3.
    Scott, R.: Sparking life: notes on the performance capture sessions for the lord of the rings: the two towers. SIGGRAPH Comput. Graph. 37(4), 17–21 (2003)CrossRefGoogle Scholar
  4. 4.
    Silver, D.: Cooperative pathfinding. In: Artificial Intelligence and Interactive Digital Entertainment Conference (2005)Google Scholar
  5. 5.
    Pelechano, N., Malkawi, A.: Evacuation simulation models: challenges in modeling high rise building evacuation with cellular automata approaches. Autom. Constr. 17(4), 377–385 (2008)CrossRefGoogle Scholar
  6. 6.
    Svestka, P., Overmars, M.H.: Coordinated path planning for multiple robots. Robot. Auton. Syst. 23(3), 125–152 (1998)CrossRefGoogle Scholar
  7. 7.
    Domke, J., Hoefler, T., Matsuoka, S.: Routing on the dependency graph: a new approach to deadlock-free high-performance routing. In: Symposium on High-Performance Parallel and Distributed Computing (2016)Google Scholar
  8. 8.
    Yu, J., Rus, D.: Pebble motion on graphs with rotations: efficient feasibility tests and planning algorithms. In: Akin, H.L., Amato, N.M., Isler, V., Stappen, A.F. (eds.) Algorithmic Foundations of Robotics XI. STAR, vol. 107, pp. 729–746. Springer, Cham (2015). doi: 10.1007/978-3-319-16595-0_42 Google Scholar
  9. 9.
    Driscoll, J.R., Furst, M.L.: On the diameter of permutation groups. In: Symposium on Theory of Computing (1983)Google Scholar
  10. 10.
    Kornhauser, D., Miller, G., Spirakis, P.: Coordinating pebble motion on graphs, the diameter of permutation groups, and applications. In: Symposium on Foundations of Computer Science (1984)Google Scholar
  11. 11.
    Johnson, W.W.: Notes on the “15” puzzle. Am. J. Math. 2(4), 397–404 (1879)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wilson, R.M.: Graph puzzles, homotopy, and the alternating group. J. Comb. Theory, Ser. B 16(1), 86–96 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ratner, D., Warmuth, M.: The \(n^2-1\) puzzle and related relocation problems. J. Symb. Comput. 10(2), 111–137 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goldreich, O.: Finding the shortest move-sequence in the graph-generalized 15-puzzle is NP-hard. In: Goldreich, O. (ed.) Studies in Complexity and Cryptography. Miscellanea on the Interplay Between Randomness and Computation. LNCS, vol. 6650, pp. 1–5. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22670-0_1 CrossRefGoogle Scholar
  15. 15.
    Kornhauser, D.: Coordinating pebble motion on graphs, the diameter of permutation groups, and applications. Master’s thesis MIT/LCS/TR-320, Massachusetts Institute of Technology (1984)Google Scholar
  16. 16.
    Röger, G., Helmert, M.: Non-optimal multi-agent pathfinding is solved (since 1984). In: Symposium on Combinatorial Search (2012)Google Scholar
  17. 17.
    Jacobson, N.: Basic Algebra. Freeman, San Francisco (1974)zbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.ETH ZurichZurichSwitzerland
  2. 2.Aalborg UniversityAalborgDenmark

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