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

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 $\mathbf{\pi}$, 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 $\mathbf{\pi}$ 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 $\epsilon > 0$, we present an algorithm for computing a coordinated motion plan for $k$ unit radius square robots of cost at most $(1+\epsilon)opt(\mathbf{s},\mathbf{t}, 1+\epsilon)+\epsilon$, which improves to $(1+\epsilon)opt(\mathbf{s},\mathbf{t}, 1+\epsilon)$ if $opt(\mathbf{s},\mathbf{t}, 1+\epsilon)\geq 1$, that runs in time $f(k,\epsilon)n^{O(k)}$, where $f(k,\epsilon) = (k/\epsilon)^{O(k^2)}$. Our result is the first polynomial-time bicriteria $(1+\epsilon)$-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.