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

2025-10-24

AI Summary

This paper addresses the minimum-sum multi-robot path planning (MPP) problem for $k$ unit-square robots navigating collision-freely from given start to goal positions in a polygonal planar environment. For constant $k > 2$, we present the first polynomial-time bi-criteria $(1+varepsilon)$-approximation algorithm: it runs in time $f(k,varepsilon) cdot n^{O(k)}$ and returns a solution whose total path length is at most $(1+varepsilon)cdot mathrm{OPT} + varepsilon$, yielding a strict $(1+varepsilon)$-approximation ratio when $mathrm{OPT} geq 1$. Our approach models the geometric configuration space and integrates obstacle-aware approximation techniques, applicable to both unit-square and congruent-disk robot models. This work overcomes a longstanding theoretical barrier—namely, the absence of efficient approximation algorithms for $k geq 3$—establishing the first provably near-optimal solution for constant-size teams in polygonal domains.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Multiagent Systems: Multiagent Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $mathbf{pi}$, 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 extit{min-sum motion planning}. Even computing a feasible motion plan for $k$ unit-square robots in a polygonal environment is { extsf 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.

Problem

• Computing collision-free coordinated paths for multiple robots in polygonal environments
• Minimizing the total path length sum for all robots' movements
• Providing near-optimal solutions for multi-robot motion planning with obstacles

Innovation

• Approximates min-sum motion planning with near-optimal cost
• Handles multiple unit-square robots in polygonal environments
• Uses bicriteria approximation for collision-free coordinated paths