Sliding Cubes in Parallel

TL;DR

This paper investigates the complexity of reconfiguring connected cube configurations in three dimensions under parallel moves, establishing NP-hardness and approximation hardness results, and proposing an input-sensitive reconfiguration algorithm.

Contribution

It generalizes known bounds from 2D to 3D, proves NP-hardness for key decision problems, and introduces an asymptotically optimal reconfiguration algorithm.

Findings

Deciding reconfiguration existence is NP-hard, even with constant makespan.

Determining if the optimal makespan is 1 or 2 is NP-hard.

The problem is log-APX-hard, indicating strong approximation difficulty.

Abstract

We study the classic sliding cube model for programmable matter under parallel reconfiguration in three dimensions, providing novel algorithmic and surprising complexity results in addition to generalizing the best known bounds from two to three dimensions. In general, the problem asks for reconfiguration sequences between two connected configurations of $n$ indistinguishable unit cube modules under connectivity constraints; a connected backbone must exist at all times. The makespan of a reconfiguration sequence is the number of parallel moves performed. We show that deciding the existence of such a sequence is NP-hard, even for constant makespan and if the two input configurations have constant-size symmetric difference, solving an open question in [Akitaya et al., ESA 25]. In particular, deciding whether the optimal makespan is 1 or 2 is NP-hard. We also show log-APX-hardness of the problem in sequential and parallel models, strengthening the APX-hardness claim in [Akitaya et al., SWAT 22]. Finally, we outline an asymptotically worst-case optimal input-sensitive algorithm for reconfiguration. The produced sequence has length that depends on the bounding box of the input configurations which, in the worst case, results in a $O(n)$ makespan.