Near-Optimal Min-Sum Motion Planning in a Planar Polygonal Environment
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arXiv
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| Format: | Preprint |
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2025
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| author | Agarwal, Pankaj K. Holmgren, Benjamin Steiger, Alex |
| author_facet | Agarwal, Pankaj K. Holmgren, Benjamin Steiger, Alex |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-Optimal Min-Sum Motion Planning in a Planar Polygonal Environment Agarwal, Pankaj K. Holmgren, Benjamin Steiger, Alex Computational Geometry 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. |
| title | Near-Optimal Min-Sum Motion Planning in a Planar Polygonal Environment |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2510.21639 |