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**1 - 3**of**3**### Navigation of Distinct Euclidean Particles via Hierarchical Clustering

"... Abstract. We present a centralized online (completely reactive) hybrid navigation algorithm for bringing a swarm of n perfectly sensed and ac-tuated point particles in Euclidean d space (for arbitrary n and d) to an arbitrary goal configuration with the guarantee of no collisions along the way. Our ..."

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Abstract. We present a centralized online (completely reactive) hybrid navigation algorithm for bringing a swarm of n perfectly sensed and ac-tuated point particles in Euclidean d space (for arbitrary n and d) to an arbitrary goal configuration with the guarantee of no collisions along the way. Our construction entails a discrete abstraction of configurations using cluster hierarchies, and relies upon two prior recent constructions: discrete dynamical system for navigating through the space of cluster hierarchies. Here, we relate the (combinatorial) topology of hierarchical clusters to the (continuous) topology of configurations by constructing “portals ” — open sets of configurations supporting two adjacent hierar-chies. The resulting online sequential composition of hierarchy-invariant swarming followed by discrete selection of a hierarchy “closer ” to that of the destination along with its continuous instantiation via an appropri-ate portal configuration yields a computationally effective construction for the desired navigation policy.

### PSPACE-hardness of unlabeled motion planning and variants

, 2014

"... In unlabeled multi-robot motion planning several interchangeable robots operate in a common workspace. The goal is to move the robots to a set of target positions such that each position will be occupied by some robot. In this paper, we study this problem for the specific case of unit-square robots ..."

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In unlabeled multi-robot motion planning several interchangeable robots operate in a common workspace. The goal is to move the robots to a set of target positions such that each position will be occupied by some robot. In this paper, we study this problem for the specific case of unit-square robots moving amidst polygonal obstacles and show that it is PSPACE-hard. We also consider three additional variants of this problem and show that they are all PSPACE-hard as well. To the best of our knowledge, this is the first hardness proof for the unlabeled case. Furthermore, our proofs can be used to show that the labeled variant (where each robot is assigned with a specific target position), again, for unit-square robots, is PSPACE-hard as well, which sets another precedence, as previous hardness results require the robots to be of different shape. 1