Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-offs

Abstract

We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Banyassady et al. (SoCG'22) guarantee feasibility in simple polygons under start--start and target--target distances of at least $4$, and start--target distances of at least $3$, but without optimality guarantees. Solovey et al. (RSS'15) provide a near-optimal solution in general polygonal domains, under stricter conditions: start/target positions must have pairwise distance at least $4$, and at least $\sqrt{5}\approx2.236$ from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is $2\tfrac{2}{3}$ and the obstacles-separation is $1\tfrac{2}{3}$, or (ii) the robots-separation is $\approx3.291$ and the obstacles-separation $\approx1.354$. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only $2$, and the obstacles-separation is $3$. Finally, we show that without any robots-separation assumption, obstacles-separation of at least $1.5$ may be necessary for a solution to exist.

Figures (13)

• Figure 1: Overlapping, blocking, and interrupting positions.
• Figure 2: A geodesic path $\gamma$ in $\mathcal{F}$ from a start position $\gamma(0)\in S$ to a target position $\gamma(1)\in T$ is illustrated in red. The obstacles are in gray, and the trace of $\gamma$ is highlighted in yellow. A position $p$ is blocking/interrupting $\gamma$. The point $a$ is the closest point to $p$ on $\gamma$.
• Figure 4: The start position $s_k$ is the last to ${\varepsilon}$-block $\gamma_i$. The switch paths are $\gamma_i'$ (dashed blue) and $\gamma_k'$ (red).
• Figure 5: Left: the robot $B$ is initially positioned at $p$. In red is a maximal subpath of $\gamma_A$ where $B$ needs to clear the way. Middle, right: the robot $B$ move inside $\mathcal{D}_{\varepsilon}(p)$ while maintaining distance exactly $2$ from the robot $A$.
• Figure 6: A instance of MRMP with $\omega\approx1.614$, in which a monotone solution does not exist.

Theorems & Definitions (18)

• Definition 1
• Lemma 1
• Definition 2
• Theorem 3
• Theorem 4
• Lemma 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8