Overview
Multi-robot motion planning requires determining collision-free paths from a source configuration $\mathbf{s}$ to a target configuration $\mathbf{t}$ within a geometric domain while optimizing for trajectory efficiency. A study detailed in arXiv:2610.20244v2 addresses this problem within a polygonal workspace $\mathcal{W}$ defined by $n$ vertices, traversed by a fixed number $k$ of translating robots $R_1, \dots, R_k$, each modeled as a polygon of constant complexity. The authors establish the first fully polynomial-time approximation schemes (FPTAS) capable of solving this problem for any fixed $k$ under three primary cost functions: min-sum, min-max, and min-makespan.
Research Context
Prior theoretical milestones achieved approximation schemes for restricted scenarios. Work by Agarwal, Halperin, Sharir, and Steiger (SODA'24) yielded an FPTAS restricted to $k = 2$ unit-square robots. Subsequently, Agarwal, Holmgren, and Steiger (SODA'26) formulated a bicriteria-approximation algorithm accommodating a fixed number $k$ of unit-square or unit-disk robots. The existence of a true FPTAS for an arbitrary fixed number of robots remained an open question raised in both studies.
Approach
The formulation analyzes translating robots modeled as constant-complexity polygons navigating an $n$-vertex polygonal environment $\mathcal{W}$. The algorithm optimizes collision-free navigation between source configuration $\mathbf{s}$ and target configuration $\mathbf{t}$ under distinct optimization targets:
- Min-sum objective: Minimizing the aggregate traveling distance accumulated across all $k$ robots.
- Min-max objective: Minimizing the maximum traveling distance traversed by any individual robot among the fleet.
- Min-makespan objective: Minimizing total operational time cost, defined under the premise that all robots operate up to a uniform maximum speed of $1$.
The authors note that the algorithmic framework is substantially simpler than previous constructions and scales geometrically: it directly generalizes to arbitrary fixed dimensions $d$, operating over polytope environments containing translating polytope robots.
Findings
The research resolves the theoretical gap left open by Agarwal et al. [SODA'24, SODA'26] by establishing:
- The first true FPTAS for motion planning involving any fixed number $k$ of translating polygonal robots in a polygonal environment.
- Simultaneous support for min-sum, min-max, and min-makespan optimization objectives.
- Direct theoretical extension to higher-dimensional spaces involving polytope robots translating through polytope environments in any fixed dimension $d$.
- A simplified design relative to earlier bicriteria and two-robot approximation methods.
Why This Matters
The result settles an established open problem in algorithmic motion planning, demonstrating that polynomial-time approximation guarantees are achievable for multiple translating polygonal bodies under diverse geometric and temporal cost metrics without resorting to bicriteria approximations.