Sliding Cubes in Parallel

TL;DR

It is shown 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].

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.

Figures (9)

• Figure 1: Two types of legal move, (i) slide and (ii) convex transition.
• Figure 2: Examples of collisions. Figure from a.akitaya_et_al:LIPIcs.ESA.2025.28, used with the authors' permission.
• Figure 7: Overview of our algorithm (not to scale). In (a)--(d), we construct a snake, which we grow to $\Theta(A+h)$ modules in (e) and use to compact the remaining configuration in (f)--(g).
• Figure 8: The cell space defined by the spine of length $16$ shown in (a). For clarity, only one half is shown. In (b), skin and interior are colored differently, and (c) illustrates sections.
• Figure 9: Structure of a snake configuration. (a) Cross-section of a valid snake configuration. (b) Two-dimensional illustration using color notation for clarity.

Theorems & Definitions (17)

• Theorem 1: $\star$
• Corollary 1: $\star$
• Corollary 2
• Theorem 3: $\star$
• Lemma 3: $\star$
• Lemma 3: $\star$
• Corollary 4
• Lemma 4: $\star$
• Lemma 4
• Lemma 4: $\star$