For any weighting between empty and loaded robot moves, single-robot tile reconfiguration is NP-hard, and disjoint start/target boxes admit a constant-factor approximation with optimal carry distance for 2-scaled shapes.
Fekete, and Aaron T
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Efficient Reconfiguration of Tile Arrangements by a Single Active Robot
For any weighting between empty and loaded robot moves, single-robot tile reconfiguration is NP-hard, and disjoint start/target boxes admit a constant-factor approximation with optimal carry distance for 2-scaled shapes.