About Ungatherability of Oblivious and Asynchronous Robots on Anonymous Rings

Abstract

We investigate on gathering of identical, memoryless, and mobile robots placed on the nodes of anonymous graphs. According to the well-known Look-Compute-Move model, robots operate in asynchronous cycles. In one cycle, a robot takes a snapshot of the current configuration (Look), decides whether to stay idle or to move to one of its neighbors (Compute), and in the latter case makes the computed move (Move). Cycles are performed asynchronously for each robot. The gathering problem asks for a strategy that brings all robots to a common node.

Several papers have been investigating the problem for various settings on ring graphs due its combinatorial relevance. However, none of the provided solutions can cope with the case of four robots, the only case still open on ring graphs, even though it is conjectured that the gathering is possible. We consider the specific cases of four robots placed on a ring of seven and nine nodes. We present an exhaustive proof about the impossibility of designing a strategy that solves the gathering in the considered setting. The proof makes use of both theoretical and computer-assisted approaches. Despite the specific cases considered, the relevance of the provided proof is twofold. On the one hand, it disproves the conjecture posed by previous works. On the other hand, it provides a new approach and new insights to the gathering problem on rings.

Work supported by the Italian Ministry of Education, University, and Research: PRIN 2010N5K7EB “ARS TechnoMedia” and PRIN 2012C4E3KT “AMANDA”, and by the National Group for Scientific Computation (GNCS-INdAM).