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Multi-Agent Path Finding For Large Agents Is Intractable

Artem Agafonov, Konstantin Yakovlev

TL;DR

This work proves that LA-MAPF, a MAPF variant that accounts for agent size in a planar embedding, is NP-hard by a polynomial reduction from 3-SAT. The reduction builds variable, clause, and blocking gadgets using disk-shaped agents with radius $r$, such that a satisfying assignment exists if and only if there is a conflict-free LA-MAPF solution for the constructed instance. It further shows how to translate between 3-SAT solutions and LA-MAPF solutions in both directions, and analyzes the implications for tractability, including discussion of assumptions and potential future work on embeddings and multi-move settings. The results justify the need for heuristics and specialized algorithms in practice and identify several avenues for future theoretical refinement, including NP-membership proofs and tractable graph families.

Abstract

The multi-agent path finding (MAPF) problem asks to find a set of paths on a graph such that when synchronously following these paths the agents never encounter a conflict. In the most widespread MAPF formulation, the so-called Classical MAPF, the agents sizes are neglected and two types of conflicts are considered: occupying the same vertex or using the same edge at the same time step. Meanwhile in numerous practical applications, e.g. in robotics, taking into account the agents' sizes is vital to ensure that the MAPF solutions can be safely executed. Introducing large agents yields an additional type of conflict arising when one agent follows an edge and its body overlaps with the body of another agent that is actually not using this same edge (e.g. staying still at some distinct vertex of the graph). Until now it was not clear how harder the problem gets when such conflicts are to be considered while planning. Specifically, it was known that Classical MAPF problem on an undirected graph can be solved in polynomial time, however no complete polynomial-time algorithm was presented to solve MAPF with large agents. In this paper we, for the first time, establish that the latter problem is NP-hard and, thus, if P!=NP no polynomial algorithm for it can, unfortunately, be presented. Our proof is based on the prevalent in the field technique of reducing the seminal 3SAT problem (which is known to be an NP-complete problem) to the problem at hand. In particular, for an arbitrary 3SAT formula we procedurally construct a dedicated graph with specific start and goal vertices and show that the given 3SAT formula is satisfiable iff the corresponding path finding instance has a solution.

Multi-Agent Path Finding For Large Agents Is Intractable

TL;DR

This work proves that LA-MAPF, a MAPF variant that accounts for agent size in a planar embedding, is NP-hard by a polynomial reduction from 3-SAT. The reduction builds variable, clause, and blocking gadgets using disk-shaped agents with radius , such that a satisfying assignment exists if and only if there is a conflict-free LA-MAPF solution for the constructed instance. It further shows how to translate between 3-SAT solutions and LA-MAPF solutions in both directions, and analyzes the implications for tractability, including discussion of assumptions and potential future work on embeddings and multi-move settings. The results justify the need for heuristics and specialized algorithms in practice and identify several avenues for future theoretical refinement, including NP-membership proofs and tractable graph families.

Abstract

The multi-agent path finding (MAPF) problem asks to find a set of paths on a graph such that when synchronously following these paths the agents never encounter a conflict. In the most widespread MAPF formulation, the so-called Classical MAPF, the agents sizes are neglected and two types of conflicts are considered: occupying the same vertex or using the same edge at the same time step. Meanwhile in numerous practical applications, e.g. in robotics, taking into account the agents' sizes is vital to ensure that the MAPF solutions can be safely executed. Introducing large agents yields an additional type of conflict arising when one agent follows an edge and its body overlaps with the body of another agent that is actually not using this same edge (e.g. staying still at some distinct vertex of the graph). Until now it was not clear how harder the problem gets when such conflicts are to be considered while planning. Specifically, it was known that Classical MAPF problem on an undirected graph can be solved in polynomial time, however no complete polynomial-time algorithm was presented to solve MAPF with large agents. In this paper we, for the first time, establish that the latter problem is NP-hard and, thus, if P!=NP no polynomial algorithm for it can, unfortunately, be presented. Our proof is based on the prevalent in the field technique of reducing the seminal 3SAT problem (which is known to be an NP-complete problem) to the problem at hand. In particular, for an arbitrary 3SAT formula we procedurally construct a dedicated graph with specific start and goal vertices and show that the given 3SAT formula is satisfiable iff the corresponding path finding instance has a solution.
Paper Structure (14 sections, 5 theorems, 17 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 5 theorems, 17 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Related works
  2. Extending MAPF To Handle Large Agents
  3. MAPF Complexity
  4. Problem Statement
  5. Reduction Of 3-SAT To LA-MAPF
  6. Variables
  7. Clauses
  8. Blocking
  9. Correctness of reduction
  10. 3-SAT to LA-MAPF
  11. LA-MAPF to 3-SAT
  12. On The Intractability Of LA-MAPF Problem
  13. Discussion
  14. Conclusions And Future Work

Key Result

Lemma 1

If there are agents in the $C_i$ and $F_j^{x_i}$ vertices (similarly for the $B_i$ and $F_j^{\neg x_i}$ vertices), then there is a conflict between them.

Figures (6)

  • Figure 1: MAPF vs. LA-MAPF.
  • Figure 2: The division of the plane for the reduction.
  • Figure 3: The gadgets of first type.
  • Figure 4: The subgraph of the clause-agent.
  • Figure 5: The subgraph of the block-agents.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof