Near-Optimal Min-Sum Motion Planning in a Planar Polygonal Environment

TL;DR

The paper presents a polynomial-time, bicriteria $(1+ε)$-approximation algorithm for min-sum motion planning of $k$ unit-square robots in a planar polygonal environment with obstacles, provided a feasible ε-robust plan exists. It constructs a finite graph in the $2k$-dimensional free space by sampling landmark-neighborhoods and proves that a near-optimal plan can be represented as a tame, low-breakpoint trajectory, enabling efficient graph search. Central to the method are two surgeries: corridor-based local parking strategies and wide-area rewiring via revolving areas, which systematically remove far breakpoints while preserving feasibility and bounding the additive cost. Extensions to disks and regular convex polygons are given, and a complete proof solidifies the near-optimality and runtime guarantees, making the approach practical for constant $k$ in complex environments.

Abstract

Let $W \subset \mathbb{R}^2$ be a planar polygonal environment with $n$ vertices, and let $[k] = \{1,\ldots,k\}$ denote $k$ unit-square robots translating in $W$. Given source and target placements $s_1, t_1, \ldots, s_k, t_k \in W$ for each robot, we wish to compute a collision-free motion plan $\mathbfπ$, i.e., a coordinated motion for each robot $i$ along a continuous path from $s_i$ to $t_i$ so that robot $i$ does not leave $W$ or collide with any other $j$. Moreover, we additionally require that $\mathbfπ$ minimizes the sum of the path lengths; this variant is known as \textit{min-sum motion planning}. Even computing a feasible motion plan for $k$ unit-square robots in a polygonal environment is {\textsf PSPACE}-hard. For $r > 0$, let $opt(\mathbf{s},\mathbf{t}, r)$ denote the cost of a min-sum motion plan for $k$ square robots of radius $r$ each from $\mathbf{s}=(s_1,\ldots,s_k)$ to $\mathbf{t}=(t_1,\ldots,t_k)$. Given a parameter $ε> 0$, we present an algorithm for computing a coordinated motion plan for $k$ unit radius square robots of cost at most $(1+ε)opt(\mathbf{s},\mathbf{t}, 1+ε)+ε$, which improves to $(1+ε)opt(\mathbf{s},\mathbf{t}, 1+ε)$ if $opt(\mathbf{s},\mathbf{t}, 1+ε)\geq 1$, that runs in time $f(k,ε)n^{O(k)}$, where $f(k,ε) = (k/ε)^{O(k^2)}$. Our result is the first polynomial-time bicriteria $(1+ε)$-approximation algorithm for any optimal multi-robot motion planning problem amidst obstacles for a constant value of $k > 2$. The algorithm also works even if robots are modeled as $k$ congruent disks.

Figures (13)

• Figure 1: (left) An example corridor $T$ containing a $\Delta$-far region, $T^S(\Delta)$. (right) A revolving area.
• Figure 2: A corridor $T$, with designated parking places defined by Lemma \ref{['lem:thin-corridor-vertices']} at the sequences of points $P^- = \langle p_1^-, \ldots, p_k^- \rangle$ and $P^+ = \langle p_1^+, \ldots, p_k^+ \rangle$. We use $P^-$ and $P^+$ to move robots across $T^S(\Delta + 10k)$ without creating breakpoints.
• Figure 3: (Left) an example configuration satisfying Lemma \ref{['lem:geodesic-plan']}. Note that the order type is defined by $\theta_{ij} \in \{x,y\}$ for each pair $(i,j) \in [4]^2$, where $\theta_{12} = x, \theta_{13} = y, \theta_{14}=y, \theta_{23}=y, \theta_{24}=x \text{ or } y, \text{ and } \theta_{34} = x$. (Right) Reducing the number of vertices in $\square_i \cap \mathcal{F}_\rho$ using $\rho$-robustness, by snapping vertices of $\mathcal{F}$ to the nearest $\rho/2$-width grid vertices contained in $\mathcal{F}_\rho$. The blue polygonal curve $P^*$ after snapping is used in place of $\partial(\mathcal{F})$. The resulting free space $\mathcal{F}^*$ now has $|V(\mathcal{F}^*_i)| \leq 2/\rho$, and still contains $\mathcal{F}_\rho$.
• Figure 4: An example decoupled $((p,a), (p,b))$-plan $\boldsymbol{\pi}$, which works identically to the one for squares in Lemma \ref{['lem:jiggle']}.
• Figure 5: A pair of octagonal robots that are $\theta = \pi/4$-separated.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3: Lemma 2.2 of steiger2024
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6