Min-Sum Uniform Coverage Problem by Autonomous Mobile Robots

We study the \textit{min-sum uniform coverage} problem for a swarm of $n$ mobile robots on a given finite line segment and on a circle having finite positive radius, where the circle is given as an input. The robots must coordinate their movements to reach a uniformly spaced configuration that minimizes the total distance traveled by all robots. The robots are autonomous, anonymous, identical, and homogeneous, and operate under the \textit{Look-Compute-Move} (LCM) model with \textit{non-rigid} motion controlled by a fair asynchronous scheduler. They are oblivious and silent, possessing neither persistent memory nor a means of explicit communication. In the \textbf{line-segment setting}, the \textit{min-sum uniform coverage} problem requires placing the robots at uniformly spaced points along the segment so as to minimize the total distance traveled by all robots. In the \textbf{circle setting} for this problem, the robots have to arrange themselves uniformly around the given circle to form a regular $n$-gon. There is no fixed orientation or designated starting vertex, and the goal is to minimize the total distance traveled by all the robots. We present a deterministic distributed algorithm that achieves uniform coverage in the line-segment setting with minimum total movement cost. For the circle setting, we characterize all initial configurations for which the \textit{min-sum uniform coverage} problem is deterministically unsolvable under the considered robot model. For all the other remaining configurations, we provide a deterministic distributed algorithm that achieves uniform coverage while minimizing the total distance traveled. These results characterize the deterministic solvability of min-sum coverage for oblivious robots and achieve optimal cost whenever solvable.

💡 Research Summary

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The paper investigates the min‑sum uniform coverage problem for a swarm of (n) autonomous mobile robots operating under the most restrictive assumptions: they are anonymous, identical, memoryless (oblivious), silent, have global visibility, and lack any common coordinate orientation or chirality. The robots follow the classic Look‑Compute‑Move (LCM) cycle in a fully asynchronous (ASYNC) scheduler, and their motion is non‑rigid, meaning an adversary may stop a robot before it reaches its computed destination, but it must advance at least a fixed positive distance (\delta) each time it moves.

Two deployment scenarios are considered:

1. Line‑segment setting – Robots are initially placed at distinct points on a finite segment (