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Optimal Path Planning in Hostile Environments

Andrzej Kaczmarczyk, Šimon Schierreich, Nicholas Axel Tanujaya, Haifeng Xu

Abstract

Coordinating agents through hazardous environments, such as aid-delivering drones navigating conflict zones or field robots traversing deployment areas filled with obstacles, poses fundamental planning challenges. We introduce and analyze the computational complexity of a new multi-agent path planning problem that captures this setting. A group of identical agents begins at a common start location and must navigate a graph-based environment to reach a common target. The graph contains hazards that eliminate agents upon contact but then enter a known cooldown period before reactivating. In this discrete-time, fully-observable, deterministic setting, the planning task is to compute a movement schedule that maximizes the number of agents reaching the target. We first prove that, despite the exponentially large space of feasible plans, optimal plans require only polynomially-many steps, establishing membership in NP. We then show that the problem is NP-hard even when the environment graph is a tree. On the positive side, we present a polynomial-time algorithm for graphs consisting of vertex-disjoint paths from start to target. Our results establish a rich computational landscape for this problem, identifying both intractable and tractable fragments.

Optimal Path Planning in Hostile Environments

Abstract

Coordinating agents through hazardous environments, such as aid-delivering drones navigating conflict zones or field robots traversing deployment areas filled with obstacles, poses fundamental planning challenges. We introduce and analyze the computational complexity of a new multi-agent path planning problem that captures this setting. A group of identical agents begins at a common start location and must navigate a graph-based environment to reach a common target. The graph contains hazards that eliminate agents upon contact but then enter a known cooldown period before reactivating. In this discrete-time, fully-observable, deterministic setting, the planning task is to compute a movement schedule that maximizes the number of agents reaching the target. We first prove that, despite the exponentially large space of feasible plans, optimal plans require only polynomially-many steps, establishing membership in NP. We then show that the problem is NP-hard even when the environment graph is a tree. On the positive side, we present a polynomial-time algorithm for graphs consisting of vertex-disjoint paths from start to target. Our results establish a rich computational landscape for this problem, identifying both intractable and tractable fragments.
Paper Structure (17 sections, 27 theorems, 6 figures, 1 algorithm)

This paper contains 17 sections, 27 theorems, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Related Work
  4. Classical Path Finding.
  5. Dynamic and Temporal Environment.
  6. Multi-Agent Path Finding.
  7. Pursuit-Evasion Games.
  8. Token Reconfiguration.
  9. Video Games and Motion Planning.
  10. Security Games.
  11. The Model
  12. The Existence of Short Plans
  13. Efficient Algorithms
  14. Limits of Tractability
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathcal{I} = (G, s, t, R, c, A, k)$ be an instance of Routing Plan and let $\rho$ be a plan of length $\ell$ with at least $k$ surviving assets witnessing that $\mathcal{I}$ is a Yes-instance. Then, there is a plan $\rho'$ of length $\ell'\in\operatorname{poly}(n,m)$ with at least $k$ survivin

Figures (6)

  • Figure 1: An illustration of our problem showing timesteps $0$ to $5$, as indicated by clock. There are two assets, red and blue. The reload time of the single trap $r$ (bomb) is $c(r) = 1$. If the trap is depicted in light gray (bomb), then it is inactive in this round.
  • Figure 2: Illustration of the construction from \ref{['thm:traps-limited-hardness']}. The dark labels by the traps (bomb) depict the respective reload times.
  • Figure 3: Construction of the reduction on trees
  • Figure 4: Variable Gadget for APX-hardness reduction.
  • Figure 5: Clause Gadget for APX-hardness reduction.
  • ...and 1 more figures

Theorems & Definitions (54)

  • Example 1
  • Example 2
  • Remark 1
  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Corollary 1
  • ...and 44 more