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 $[k]=\{1,\dots ,k\}$ denote $k$ unit-square robots translating in $W$. Given source and target placements $s_1,t_1,\dots ,s_k,t_k\in W$ 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 $\mathit{\text{min-sum}}\mathit{\text{motion}}\mathit{\text{planning}}$. 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 $\mathbf{s}=(s_1,\dots ,s_k)$ to $\mathbf{t}=(t_1,\dots ,t_k)$. 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 $opt(\mathbf{s},\mathbf{t},1+\epsilon )\ge 1$, that runs in time $f(k,\epsilon )n^{O(k)}$, where $f(k,\epsilon )=(k\mathrm{/}\epsilon {)}^{O(k^2)}$. 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.